  Raffi Hovasapian

Double Integrals

Slide Duration:

Section 1: Vectors
Points & Vectors

28m 23s

Intro
0:00
Points and Vectors
1:02
A Point in a Plane
1:03
A Point in Space
3:14
Notation for a Space of a Given Space
6:34
Introduction to Vectors
9:51
14:51
Example 1
16:52
18:24
Example 2
21:01
Two More Properties of Vector Addition
24:16
Multiplication of a Vector by a Constant
25:27
Scalar Product & Norm

30m 25s

Intro
0:00
Scalar Product and Norm
1:05
Introduction to Scalar Product
1:06
Example 1
3:21
Properties of Scalar Product
6:14
Definition: Orthogonal
11:41
Example 2: Orthogonal
14:19
Definition: Norm of a Vector
15:30
Example 3
19:37
Distance Between Two Vectors
22:05
Example 4
27:19
More on Vectors & Norms

38m 18s

Intro
0:00
More on Vectors and Norms
0:38
Open Disc
0:39
Close Disc
3:14
Open Ball, Closed Ball, and the Sphere
5:22
Property and Definition of Unit Vector
7:16
Example 1
14:04
Three Special Unit Vectors
17:24
General Pythagorean Theorem
19:44
Projection
23:00
Example 2
28:35
Example 3
35:54
Inequalities & Parametric Lines

33m 19s

Intro
0:00
Inequalities and Parametric Lines
0:30
Starting Example
0:31
Theorem 1
5:10
Theorem 2
7:22
Definition 1: Parametric Equation of a Straight Line
10:16
Definition 2
17:38
Example 1
21:19
Example 2
25:20
Planes

29m 59s

Intro
0:00
Planes
0:18
Definition 1
0:19
Example 1
7:04
Example 2
12:45
General Definitions and Properties: 2 Vectors are Said to Be Paralleled If
14:50
Example 3
16:44
Example 4
20:17
More on Planes

34m 18s

Intro
0:00
More on Planes
0:25
Example 1
0:26
Distance From Some Point in Space to a Given Plane: Derivation
10:12
Final Formula for Distance
21:20
Example 2
23:09
Example 3: Part 1
26:56
Example 3: Part 2
31:46
Section 2: Differentiation of Vectors
Maps, Curves & Parameterizations

29m 48s

Intro
0:00
Maps, Curves and Parameterizations
1:10
Recall
1:11
Looking at y = x2 or f(x) = x2
2:23
Departure Space & Arrival Space
7:01
Looking at a 'Function' from ℝ to ℝ2
10:36
Example 1
14:50
Definition 1: Parameterized Curve
17:33
Example 2
21:56
Example 3
25:16
Differentiation of Vectors

39m 40s

Intro
0:00
Differentiation of Vectors
0:18
Example 1
0:19
Definition 1: Velocity of a Curve
1:45
Line Tangent to a Curve
6:10
Example 2
7:40
Definition 2: Speed of a Curve
12:18
Example 3
13:53
Definition 3: Acceleration Vector
16:37
Two Definitions for the Scalar Part of Acceleration
17:22
Rules for Differentiating Vectors: 1
19:52
Rules for Differentiating Vectors: 2
21:28
Rules for Differentiating Vectors: 3
22:03
Rules for Differentiating Vectors: 4
24:14
Example 4
26:57
Section 3: Functions of Several Variables
Functions of Several Variable

29m 31s

Intro
0:00
Length of a Curve in Space
0:25
Definition 1: Length of a Curve in Space
0:26
Extended Form
2:06
Example 1
3:40
Example 2
6:28
Functions of Several Variable
8:55
Functions of Several Variable
8:56
General Examples
11:11
Graph by Plotting
13:00
Example 1
16:31
Definition 1
18:33
Example 2
22:15
Equipotential Surfaces
25:27
Isothermal Surfaces
27:30
Partial Derivatives

23m 31s

Intro
0:00
Partial Derivatives
0:19
Example 1
0:20
Example 2
5:30
Example 3
7:48
Example 4
9:19
Definition 1
12:19
Example 5
14:24
Example 6
16:14
20:26
Higher and Mixed Partial Derivatives

30m 48s

Intro
0:00
Higher and Mixed Partial Derivatives
0:45
Definition 1: Open Set
0:46
Notation: Partial Derivatives
5:39
Example 1
12:00
Theorem 1
14:25
Now Consider a Function of Three Variables
16:50
Example 2
20:09
Caution
23:16
Example 3
25:42
Section 4: Chain Rule and The Gradient
The Chain Rule

28m 3s

Intro
0:00
The Chain Rule
0:45
Conceptual Example
0:46
Example 1
5:10
The Chain Rule
10:11
Example 2: Part 1
19:06
Example 2: Part 2 - Solving Directly
25:26
Tangent Plane

42m 25s

Intro
0:00
Tangent Plane
1:02
Tangent Plane Part 1
1:03
Tangent Plane Part 2
10:00
Tangent Plane Part 3
18:18
Tangent Plane Part 4
21:18
Definition 1: Tangent Plane to a Surface
27:46
Example 1: Find the Equation of the Plane Tangent to the Surface
31:18
Example 2: Find the Tangent Line to the Curve
36:54
Further Examples with Gradients & Tangents

47m 11s

Intro
0:00
Example 1: Parametric Equation for the Line Tangent to the Curve of Two Intersecting Surfaces
0:41
Part 1: Question
0:42
Part 2: When Two Surfaces in ℝ3 Intersect
4:31
Part 3: Diagrams
7:36
Part 4: Solution
12:10
Part 5: Diagram of Final Answer
23:52
Example 2: Gradients & Composite Functions
26:42
Part 1: Question
26:43
Part 2: Solution
29:21
Example 3: Cos of the Angle Between the Surfaces
39:20
Part 1: Question
39:21
Part 2: Definition of Angle Between Two Surfaces
41:04
Part 3: Solution
42:39
Directional Derivative

41m 22s

Intro
0:00
Directional Derivative
0:10
Rate of Change & Direction Overview
0:11
Rate of Change : Function of Two Variables
4:32
Directional Derivative
10:13
Example 1
18:26
Examining Gradient of f(p) ∙ A When A is a Unit Vector
25:30
Directional Derivative of f(p)
31:03
33:23
Example 2
34:53
A Unified View of Derivatives for Mappings

39m 41s

Intro
0:00
A Unified View of Derivatives for Mappings
1:29
Derivatives for Mappings
1:30
Example 1
5:46
Example 2
8:25
Example 3
12:08
Example 4
14:35
Derivative for Mappings of Composite Function
17:47
Example 5
22:15
Example 6
28:42
Section 5: Maxima and Minima
Maxima & Minima

36m 41s

Intro
0:00
Maxima and Minima
0:35
Definition 1: Critical Point
0:36
Example 1: Find the Critical Values
2:48
Definition 2: Local Max & Local Min
10:03
Theorem 1
14:10
Example 2: Local Max, Min, and Extreme
18:28
Definition 3: Boundary Point
27:00
Definition 4: Closed Set
29:50
Definition 5: Bounded Set
31:32
Theorem 2
33:34
Further Examples with Extrema

32m 48s

Intro
0:00
Further Example with Extrema
1:02
Example 1: Max and Min Values of f on the Square
1:03
Example 2: Find the Extreme for f(x,y) = x² + 2y² - x
10:44
Example 3: Max and Min Value of f(x,y) = (x²+ y²)⁻¹ in the Region (x -2)²+ y² ≤ 1
17:20
Lagrange Multipliers

32m 32s

Intro
0:00
Lagrange Multipliers
1:13
Theorem 1
1:14
Method
6:35
Example 1: Find the Largest and Smallest Values that f Achieves Subject to g
9:14
Example 2: Find the Max & Min Values of f(x,y)= 3x + 4y on the Circle x² + y² = 1
22:18
More Lagrange Multiplier Examples

27m 42s

Intro
0:00
Example 1: Find the Point on the Surface z² -xy = 1 Closet to the Origin
0:54
Part 1
0:55
Part 2
7:37
Part 3
10:44
Example 2: Find the Max & Min of f(x,y) = x² + 2y - x on the Closed Disc of Radius 1 Centered at the Origin
16:05
Part 1
16:06
Part 2
19:33
Part 3
23:17
Lagrange Multipliers, Continued

31m 47s

Intro
0:00
Lagrange Multipliers
0:42
First Example of Lesson 20
0:44
Let's Look at This Geometrically
3:12
Example 1: Lagrange Multiplier Problem with 2 Constraints
8:42
Part 1: Question
8:43
Part 2: What We Have to Solve
15:13
Part 3: Case 1
20:49
Part 4: Case 2
22:59
Part 5: Final Solution
25:45
Section 6: Line Integrals and Potential Functions
Line Integrals

36m 8s

Intro
0:00
Line Integrals
0:18
Introduction to Line Integrals
0:19
Definition 1: Vector Field
3:57
Example 1
5:46
Example 2: Gradient Operator & Vector Field
8:06
Example 3
12:19
Vector Field, Curve in Space & Line Integrals
14:07
Definition 2: F(C(t)) ∙ C'(t) is a Function of t
17:45
Example 4
18:10
Definition 3: Line Integrals
20:21
Example 5
25:00
Example 6
30:33
More on Line Integrals

28m 4s

Intro
0:00
More on Line Integrals
0:10
Line Integrals Notation
0:11
Curve Given in Non-parameterized Way: In General
4:34
Curve Given in Non-parameterized Way: For the Circle of Radius r
6:07
Curve Given in Non-parameterized Way: For a Straight Line Segment Between P & Q
6:32
The Integral is Independent of the Parameterization Chosen
7:17
Example 1: Find the Integral on the Ellipse Centered at the Origin
9:18
Example 2: Find the Integral of the Vector Field
16:26
Discussion of Result and Vector Field for Example 2
23:52
Graphical Example
26:03
Line Integrals, Part 3

29m 30s

Intro
0:00
Line Integrals
0:12
Piecewise Continuous Path
0:13
Closed Path
1:47
Example 1: Find the Integral
3:50
The Reverse Path
14:14
Theorem 1
16:18
Parameterization for the Reverse Path
17:24
Example 2
18:50
Line Integrals of Functions on ℝn
21:36
Example 3
24:20
Potential Functions

40m 19s

Intro
0:00
Potential Functions
0:08
Definition 1: Potential Functions
0:09
Definition 2: An Open Set S is Called Connected if…
5:52
Theorem 1
8:19
Existence of a Potential Function
11:04
Theorem 2
18:06
Example 1
22:18
Contrapositive and Positive Form of the Theorem
28:02
The Converse is Not Generally True
30:59
Our Theorem
32:55
Compare the n-th Term Test for Divergence of an Infinite Series
36:00
So for Our Theorem
38:16
Potential Functions, Continued

31m 45s

Intro
0:00
Potential Functions
0:52
Theorem 1
0:53
Example 1
4:00
Theorem in 3-Space
14:07
Example 2
17:53
Example 3
24:07
Potential Functions, Conclusion & Summary

28m 22s

Intro
0:00
Potential Functions
0:16
Theorem 1
0:17
In Other Words
3:25
Corollary
5:22
Example 1
7:45
Theorem 2
11:34
Summary on Potential Functions 1
15:32
Summary on Potential Functions 2
17:26
Summary on Potential Functions 3
18:43
Case 1
19:24
Case 2
20:48
Case 3
21:35
Example 2
23:59
Section 7: Double Integrals
Double Integrals

29m 46s

Intro
0:00
Double Integrals
0:52
Introduction to Double Integrals
0:53
Function with Two Variables
3:39
Example 1: Find the Integral of xy³ over the Region x ϵ[1,2] & y ϵ[4,6]
9:42
Example 2: f(x,y) = x²y & R be the Region Such That x ϵ[2,3] & x² ≤ y ≤ x³
15:07
Example 3: f(x,y) = 4xy over the Region Bounded by y= 0, y= x, and y= -x+3
19:20
Polar Coordinates

36m 17s

Intro
0:00
Polar Coordinates
0:50
Polar Coordinates
0:51
Example 1: Let (x,y) = (6,√6), Convert to Polar Coordinates
3:24
Example 2: Express the Circle (x-2)² + y² = 4 in Polar Form.
5:46
Graphing Function in Polar Form.
10:02
Converting a Region in the xy-plane to Polar Coordinates
14:14
Example 3: Find the Integral over the Region Bounded by the Semicircle
20:06
Example 4: Find the Integral over the Region
27:57
Example 5: Find the Integral of f(x,y) = x² over the Region Contained by r= 1 - cosθ
32:55
Green's Theorem

38m 1s

Intro
0:00
Green's Theorem
0:38
Introduction to Green's Theorem and Notations
0:39
Green's Theorem
3:17
Example 1: Find the Integral of the Vector Field around the Ellipse
8:30
Verifying Green's Theorem with Example 1
15:35
A More General Version of Green's Theorem
20:03
Example 2
22:59
Example 3
26:30
Example 4
32:05
Divergence & Curl of a Vector Field

37m 16s

Intro
0:00
Divergence & Curl of a Vector Field
0:18
Definitions: Divergence(F) & Curl(F)
0:19
Example 1: Evaluate Divergence(F) and Curl(F)
3:43
Properties of Divergence
9:24
Properties of Curl
12:24
Two Versions of Green's Theorem: Circulation - Curl
17:46
Two Versions of Green's Theorem: Flux Divergence
19:09
Circulation-Curl Part 1
20:08
Circulation-Curl Part 2
28:29
Example 2
32:06
Divergence & Curl, Continued

33m 7s

Intro
0:00
Divergence & Curl, Continued
0:24
Divergence Part 1
0:25
Divergence Part 2: Right Normal Vector and Left Normal Vector
5:28
Divergence Part 3
9:09
Divergence Part 4
13:51
Divergence Part 5
19:19
Example 1
23:40
Final Comments on Divergence & Curl

16m 49s

Intro
0:00
Final Comments on Divergence and Curl
0:37
Several Symbolic Representations for Green's Theorem
0:38
Circulation-Curl
9:44
Flux Divergence
11:02
Closing Comments on Divergence and Curl
15:04
Section 8: Triple Integrals
Triple Integrals

27m 24s

Intro
0:00
Triple Integrals
0:21
Example 1
2:01
Example 2
9:42
Example 3
15:25
Example 4
20:54
Cylindrical & Spherical Coordinates

35m 33s

Intro
0:00
Cylindrical and Spherical Coordinates
0:42
Cylindrical Coordinates
0:43
When Integrating Over a Region in 3-space, Upon Transformation the Triple Integral Becomes..
4:29
Example 1
6:27
The Cartesian Integral
15:00
Introduction to Spherical Coordinates
19:44
Reason It's Called Spherical Coordinates
22:49
Spherical Transformation
26:12
Example 2
29:23
Section 9: Surface Integrals and Stokes' Theorem
Parameterizing Surfaces & Cross Product

41m 29s

Intro
0:00
Parameterizing Surfaces
0:40
Describing a Line or a Curve Parametrically
0:41
Describing a Line or a Curve Parametrically: Example
1:52
Describing a Surface Parametrically
2:58
Describing a Surface Parametrically: Example
5:30
Recall: Parameterizations are not Unique
7:18
Example 1: Sphere of Radius R
8:22
Example 2: Another P for the Sphere of Radius R
10:52
This is True in General
13:35
Example 3: Paraboloid
15:05
Example 4: A Surface of Revolution around z-axis
18:10
Cross Product
23:15
Defining Cross Product
23:16
Example 5: Part 1
28:04
Example 5: Part 2 - Right Hand Rule
32:31
Example 6
37:20
Tangent Plane & Normal Vector to a Surface

37m 6s

Intro
0:00
Tangent Plane and Normal Vector to a Surface
0:35
Tangent Plane and Normal Vector to a Surface Part 1
0:36
Tangent Plane and Normal Vector to a Surface Part 2
5:22
Tangent Plane and Normal Vector to a Surface Part 3
13:42
Example 1: Question & Solution
17:59
Example 1: Illustrative Explanation of the Solution
28:37
Example 2: Question & Solution
30:55
Example 2: Illustrative Explanation of the Solution
35:10
Surface Area

32m 48s

Intro
0:00
Surface Area
0:27
Introduction to Surface Area
0:28
Given a Surface in 3-space and a Parameterization P
3:31
Defining Surface Area
7:46
Curve Length
10:52
Example 1: Find the Are of a Sphere of Radius R
15:03
Example 2: Find the Area of the Paraboloid z= x² + y² for 0 ≤ z ≤ 5
19:10
Example 2: Writing the Answer in Polar Coordinates
28:07
Surface Integrals

46m 52s

Intro
0:00
Surface Integrals
0:25
Introduction to Surface Integrals
0:26
General Integral for Surface Are of Any Parameterization
3:03
Integral of a Function Over a Surface
4:47
Example 1
9:53
Integral of a Vector Field Over a Surface
17:20
Example 2
22:15
Side Note: Be Very Careful
28:58
Example 3
30:42
Summary
43:57
Divergence & Curl in 3-Space

23m 40s

Intro
0:00
Divergence and Curl in 3-Space
0:26
Introduction to Divergence and Curl in 3-Space
0:27
Define: Divergence of F
2:50
Define: Curl of F
4:12
The Del Operator
6:25
Symbolically: Div(F)
9:03
Symbolically: Curl(F)
10:50
Example 1
14:07
Example 2
18:01
Divergence Theorem in 3-Space

34m 12s

Intro
0:00
Divergence Theorem in 3-Space
0:36
Green's Flux-Divergence
0:37
Divergence Theorem in 3-Space
3:34
Note: Closed Surface
6:43
Figure: Paraboloid
8:44
Example 1
12:13
Example 2
18:50
Recap for Surfaces: Introduction
27:50
Recap for Surfaces: Surface Area
29:16
Recap for Surfaces: Surface Integral of a Function
29:50
Recap for Surfaces: Surface Integral of a Vector Field
30:39
Recap for Surfaces: Divergence Theorem
32:32
Stokes' Theorem, Part 1

22m 1s

Intro
0:00
Stokes' Theorem
0:25
Recall Circulation-Curl Version of Green's Theorem
0:26
Constructing a Surface in 3-Space
2:26
Stokes' Theorem
5:34
Note on Curve and Vector Field in 3-Space
9:50
Example 1: Find the Circulation of F around the Curve
12:40
Part 1: Question
12:48
Part 2: Drawing the Figure
13:56
Part 3: Solution
16:08
Stokes' Theorem, Part 2

20m 32s

Intro
0:00
Example 1: Calculate the Boundary of the Surface and the Circulation of F around this Boundary
0:30
Part 1: Question
0:31
Part 2: Drawing the Figure
2:02
Part 3: Solution
5:24
Example 2: Calculate the Boundary of the Surface and the Circulation of F around this Boundary
13:11
Part 1: Question
13:12
Part 2: Solution
13:56

• ## Related Books 1 answer Last reply by: Professor HovasapianFri Aug 22, 2014 8:27 PMPost by Denny Yang â™• [Moderator] on August 19, 2014Also in example 3, you made a minor calculation error. You wrote: The  integral of(2x^3 - 12x^2 +19x, x, (3/2), 3 ). It should have been (2x^3 - 12x^2 +18x, x, (3/2), 3 ). The [ x, (3/2), 3 ] is the variable we are integrating and the lower bound and upper bound. Typing it just like you would on a Ti-89 Calculator. 1 answer Last reply by: Professor HovasapianFri Aug 22, 2014 8:23 PMPost by Denny Yang â™• [Moderator] on August 19, 2014Q. ii) Integrate âˆ« âˆ’ Ï€\protect\mathord/ \protect phantom Ï€2 / 20 ey dx.What is this? 1 answer Last reply by: Professor HovasapianMon May 13, 2013 12:53 AMPost by Josh Winfield on May 12, 2013Hello Raffi,Can you just clarify why in example 3 you put the limit of integration for dx as (3-y) on the bottom and (y) on the top. I had them the other way around but im wrong but i cant figure out why? 0 answersPost by Justin Dehorty on February 23, 2013How do I find the Lower and Upper Sums? Apparently this is another way to calculate double integrals, but I don't understand how to set up the sigmas properly. 1 answer Last reply by: Professor HovasapianSat Dec 29, 2012 5:01 PMPost by Alexander Rekitsanov Alexander Rekitsanov on December 15, 2012I think that the region the prof is integrating is wrong in example 3. in the question it is stating that region that is bounded by y=0 ,y=x and y=-x+3 but the prof ignores the y=0 part. isn't the region suppose to be the left triangle in the drawing ? please if you could clarify it, thank you.

### Double Integrals

Find intervals of integration for dydx over the shaded region and set up the double integral. • To set up the integral pay close attendtion to the order of the differentials, dy and dx.
• Since dy is prior to dx we set up the integral so that we first cover the intervals for y and then those for x. That is, the intervals of y are a function of x.
• Our shaded region is the area between the curves y = x2 and y = 1. Hence x2 ≤ y ≤ 1. Note that this area runs from - 1 ≤ x ≤ 1.
Thus our intervals of integration yield the double integral ∫ − 11x21 f dydx. Note the order of integration corresponds to the opposite order of the differentials.
Find intervals of integration for dxdy over the shaded region and set up the double integral. • To set up the integral pay close attendtion to the order of the differentials, dy and dx.
• Since dx is prior to dy we set up the integral so that we first cover the intervals for x and then those for y. That is, the intervals of x are a function of y.
• Our shaded region is the left side of the circle x2 + y2 = 1. Solving for x yields x = ±√{1 − y2} , we only want the left side, so we take the negative result.
• Hence − √{1 − y2} ≤ x ≤ 0 and this area runs from − 1 ≤ y ≤ 1.
Thus our intervals of integration yield the double integral ∫ − 11 − √{1 − y2}0 f dxdy. Note the order of integration corresponds to the opposite order of the differentials.
i) Integrate ∫ − 21 5xy dy.
• Taking the integral in respect to y takes any other variable as a constant.
So ∫ − 21 5xy dy = [5/2]xy2 | − 21 = [5/2]x(1)2 − [5/2]x( − 2)2 = [5/2]x − 10x = − [15/2]x.
ii) Integrate ∫13 5xy dx.
• Taking the integral in respect to x takes any other variable as a constant.
So ∫13 5xy dx = [5/2]x2y |13 = [5/2](3)2y − [5/2](1)2y = [45/2]y − [5/2]y = 20x
iii) Integrate ∫13 − 21 5xy dydx.
• We take the integral in respect to the order of the differentials.
So ∫13 − 21 5xy dydx = ∫13− [15/2]x dx = − [15/4]x2 |13 = − [15/4]( 9 − 1 ) = − 30
iv) Integrate ∫ − 2113 5xy dxdy.
• We take the integral in respect to the order of the differentials.
So ∫ − 2113 5xy dxdy = ∫ − 21 20x dy = 10x2 | − 21 = 10( 1 − 4 ) = − 30. Note that ∫13 − 21 5xy dydx = ∫ − 2113 5xy dxdy.
i) Integrate ∫ − 20 cos(x) dy.
• Taking the integral in respect to y takes any other variable as a constant.
So ∫ − 20 cos(x) dy = |ycos(x) | − 20 = (0)(cos(x)) − ( − 2)cos(x) = 2cos(x)
ii) Integrate ∫ − π\protect\mathord/ \protect phantom π2/ 20 ey dx.
• Taking the integral in respect to x takes any other variable as a constant.
So ∫ − π\mathord/ \protect phantom π2 20 ey dx = xey | − π\mathord/ \protect phantom π2 20 = (0)ey − ( − [(π)/2] )ey = [(πey)/2]
iii) Integrate ∫ − π\mathord/ \protect phantom π2 20 − 20 eycos(x) dydx.
• We take the integral in respect to the order of the differentials.
So ∫ − π\mathord/ \protect phantom π2 20 − 20 eycos(x) dydx = ∫ − π\mathord/ \protect phantom π2 20 ( eycos(x) | − 20 ) dx = ∫ − π\mathord/ \protect phantom π2 20 ( cos(x) − e − 2cos(x) ) dx = sin(x) | − π\mathord/ \protect phantom π2 20 − e − 2sin(x) | − π\mathord/ \protect phantom π2 20 = 1 − e − 2
iv) Integrate ∫ − 20 − π\mathord/ \protect phantom π2 20 eycos(x) dxdy.
• We take the integral in respect to the order of the differentials.
So ∫ − 20 − π\mathord/ \protect phantom π2 20 eycos(x) dxdy = ∫ − 20 ( eysin(x) | − π\mathord/ \protect phantom π2 20 ) dy = ∫ − 20 ey dy = ey | − 20 = 1 − e − 2.  Note that ∫ − π\mathord/ \protect phantom π2 20 − 20 eycos(x) dydx = ∫ − 20 − π\mathord/ \protect phantom π2 20 eycos(x) dxdy.
Integrate given that y ∈ [ − 1,1], x ∈ [0,2] and f(x,y) = [1/2]xy3.
• We set up our double integral in respect to the differentials dy and dx. Note that we have dydx.
• So our double integral has the intervals of x followed by those of y. Then
Integrating yields ∫02 − 11 [1/2]xy3 dydx = ∫02 ( [1/8]xy4 | − 11 ) dx = ∫02 0dx = 0
Integrate given that y ∈ [0,π], x ∈ [ − π,π] and g(x,y) = cos(x)sin(y).
• We set up our double integral in respect to the differentials dy and dx. Note that we have dydx.
• So our double integral has the intervals of x followed by those of y. Then
Integrating yields ∫ − ππ0π cos(x)sin(y) dydx = ∫ − ππ ( − cos(x)cos(y) |0π ) dx = ∫ − ππ 2cos(x)dx = 2sin(x) | − ππ = 0
Integrate given that x ∈ [ − [1/2],[1/2] ], y ∈ [0,5] and h(x,y) = yex.
• We set up our double integral in respect to the differentials dy and dx. Note that we have dydx.
• So our double integral has the intervals of x followed by those of y. Then
Integrating yields ∫ − 1 \mathord/ \protect phantom 1 2 21 \mathord/ \protect phantom 1 2 205 yex dydx = ∫ − 1 \mathord/ \protect phantom 1 2 21 \mathord/ \protect phantom 1 2 2 ( [1/2]y2ex |05 ) dx = ∫ − 1 \mathord/ \protect phantom 1 2 21 \mathord/ \protect phantom 1 2 2 ( [25/2]ex )dx = [25/2]ex | − 1 \mathord/ \protect phantom 1 2 21 \mathord/ \protect phantom 1 2 2 = [25/2]( e1 \mathord/ \protect phantom 1 2 2 − e − 1 \mathord/ \protect phantom 1 2 2 )
Let f(x,y) = [y/(x2)] and R be the shaded region described by: Find dA.
• Note that dA is the area covered by the differentials dx and dy. We shall integrate in respect to that order, that is dxdy.
• Our region is the area between the curve y = x3 and y = x. The intervals of integration for y ∈ [x3,x].
• To find the intervals of integration for x, we note that x runs from 0 to 1 along our shaded region and so x ∈ [0,1].
Then dA = ∫01x3x [y/(x2)]dydx = ∫01 ( [(y2)/(2x2)] |x3x ) dx = ∫01 ( [((x)2)/(2x2)] − [((x3)2)/(2x2)] ) dx = ∫01 ( [1/2] − [(x4)/2] )dx = ( [1/2]x − [1/10]x5 ) |01 = [2/5]
Let f(x,y) = k where k a constant, k > 0 and R be the shaded region described by: Find dA.
• Note that dA is the area covered by the differentials dx and dy. We shall integrate in respect to that order, that is dxdy.
• Our region is the area between the curve y = 0 and y = sin(x). The intervals of integration for y ∈ [0,sin(x)].
• To find the intervals of integration for x, we note that x runs from 0 to p along our shaded region and so x ∈ [0,p].
Then dA = ∫0p0sin(x) kdydx = ∫0p ( ky |0sin(x) ) dx = ∫0p ( ksin(x) − ksin(0) ) dx = ∫0p ksin(x)dx = − kcos(x) |0p = 2k
Let f(x,y) = − 2x3y2 and R be the region bounded by x = 0, y = 1 and y = x.
i) Find dydx. Do not integrate.
• Sketching our region yields
• • Note the blue line demonstrating our intervals of integration for y, that is y ∈ [x,1]. We can also see that x ∈ [0,1].
Hence dydx = ∫01x1− 2x3y2 dydx.
Let f(x,y) = − 2x3y2 and R be the region bounded by x = 0, y = 1 and y = x.
ii) Find dxdy. Do not integrate.
• Sketching our region yields
• • Note the blue line demonstrating our intervals of integration for x, that is x ∈ [0,y]. We can also see that y ∈ [0,1].
Hence dydx = ∫010y− 2x3y2 dxdy.
Let f(x,y) = − 2x3y2 and R be the region bounded by x = 0, y = 1 and y = x.
iii) Verify that ∫01x1− 2x3y2 dydx = ∫010y− 2x3y2 dxdy .
• We integrate each side of the equation and see if they are equal.
• First ∫01x1− 2x3y2 dydx = ∫01 ( − [2/3]x3y3 |x1 )dx = ∫01 ( − [2/3]x3 + [2/3]x6 ) dx = ( − [1/6]x4 + [2/21]x7 ) |01 = − [1/14]
• Second ∫010y− 2x3y2 dxdy = ∫01 ( − [1/2]x4y2 |0y )dy = ∫01 ( − [1/2]y6 ) dy = − [1/14]y7 |01 = − [1/14]
Since both double integrals yielded the same value, then ∫01x1− 2x3y2 dydx = ∫010y− 2x3y2 dxdy . Note that the second double integral was simpler to compute.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Double Integrals

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Double Integrals 0:52
• Introduction to Double Integrals
• Function with Two Variables
• Example 1: Find the Integral of xy³ over the Region x ϵ[1,2] & y ϵ[4,6]
• Example 2: f(x,y) = x²y & R be the Region Such That x ϵ[2,3] & x² ≤ y ≤ x³
• Example 3: f(x,y) = 4xy over the Region Bounded by y= 0, y= x, and y= -x+3

### Transcription: Double Integrals

Hello and welcome back to educator.com and multivariable calculus.0000

Today we are going to start on a new topic. We are going to be talking about double integrals.0004

There is going to be some new notation, but in face the notation itself is not even new. It is exactly the same as what you saw in single variable calculus, just picked up from another dimension -- dimension 1, dimension 2.0012

So for all practical purposes, you have done this stuff that we are going to be doing over and over and over again.0025

Rather than belabor the point with a lot of theory, I am actually just going to give a couple of quick definitions, and then we are just going to launch right into examples.0032

We just want to be able to handle these problems and not worry about what is going on behind the scenes too much.0041

You already know a lot of what is going on. Okay. Let us just jump right on in.0048

Okay. Now, in single variable calculus what we did was we defined this integral of f(x)/some interval from a to b.0054

We had something that looked like this... the integral from a to b of f(x) dx, and this was some number and we learned a bunch of techniques for evaluating this.0063

Well, let us go ahead and talk about what some of these things mean.0077

This right here is just a value of f at a given x on the particular interval, and this ab is of course the closed interval from a to b, just something like this.0080

If there is some function, this is a, this is b, this is the function, so we are just integrating along that length. That is the important part, we are just integrating along a length.0095

Now, this dx here it is just a differential length element -- a differential is just a fancy word for small -- so differential length element, and this symbol right here... that is just the symbol for an infinite sum.0105

So, what we have done is we have a particular function, we have a bunch of x's in between a and b, we evaluate them at each of those x's, and then we multiply by the actual length, some length element along x and then we take these numbers that we get and we just add them up with this fancy technique called integration.0137

Then we get some number. That is the integral, that is all we have done.0159

Well, we can do the same thing with a function of two variables, except now, a function of 2 variables is not defined just over the x axis, it is defined over x and y.0162

So, instead of a differential length element, what you are going to have is a differential area element.0173

Because again, now you are going to have an interval from a to b along x, but you are also going to have an interval c to d along y, so now we are going to basically break up the a to b the way we did before in single variable calculus, but we are also going to break up c to d and what you end up having is a bunch of little rectangles.0180

Instead of integrating over a length, we are integrating over a length and another length.0205

Well, length × length is area, so we are integrating over the entire area. Everything else is exactly the same, the notation is exactly the same.0212

Let us go ahead and write this out. So, for a function of two variables, f(x,y), we can integrate this function over an area, which is just length × length.0220

You will see in a minute that is exactly what you are going to be doing. You are just going to be doing one integral at a time, in an iterated fashion, in a row.0272

The symbol for it is exactly the same, except you are going to use two integral signs instead of one.0282

So, double integral, f(x,y) dy dx.0288

Also written as double integral of f, I will leave off the x,y, da.0299

dy × dx, well, differential can be x, differential in the y-direction, that gives me a little bit of a square, so this square is a differential area element. That is why we have da for dy/dx, that is it.0310

This is the one that we want to concentrate on, but you will see this when you see certain theorems, certain statements along as you understand what is happening.0327

This is a symbolic definition, so let us talk about what these mean. This is the value of f at the point (x,y).0337

This is a differential area element.0349

So, instead of having a very, very, tiny length, like single variable calculus, what we have is a really, really tiny area. One of these small rectangles or squares.0356

This, of course, is the symbol for the infinite sum -- the symbol for infinite sum -- and we use two of them to differentiate from the single variable because we are talking about two variables.0367

Later, when we do functions of three variables, you are going to see triple integral, three integral signs. It actually exists that you can keep going.0382

Okay, that is it. That is literally all that is going on here.0391

Let me go ahead and draw out one more time this thing just to make sure we completely understand.0396

So, now, because our domain is now in two dimensions, we are going to split up the x length, we are going to split up the y length, and we are just going to add the value of the function over all of these little rectangles, that is what this is.0405

Now, I am going to write out how we actually evaluate this. The practical way of finding the integral, and we do it with a single integral at a time.0423

So, let us go here. The double integral of f(x,y) dy/dx, is evaluated as -- let me go ahead and write over here -- so from a to b, from c to d, so, our interval along the x-axis is a to b, our interval along the y axis is c to d.0434

It is equivalent to taking the integral from a to b, taking the integral from c to d of f(x,y) dy, and then dx.0478

What this symbol means is that we are going to be working from the inside out, just like we do in mathematics.0495

What we are going to do is we are going back to one variable first, in this particular case, dy, f(x,y) dy, and when we do this we have to remember that y is the variable we are integrating with respect to so we keep x constant.0500

When we do this, and we evaluate this integral from c to d, we are going to get a function x.0515

Now this function of x is what we integrate from a to b and we get our final answer. That is it, this is called iterated integration and you can actually do the integral in either order.0519

You can either do f(x,y) dy dx, or you can do f(x,y) dx dy.0531

The problem itself, the particular domain you are going to be working with, will decide for you which you are going to integrate first.0536

Again, you are just integrating one variable at a time. let us just go ahead and jump into some examples, and hopefully it will all make sense.0544

Again, you have done this over and over and over again. Now instead of just stopping after one integral, you are doing another integral after that. The only thing that you have to watch out for is keeping track of the variables.0553

If you are going to be integrating with respect to y, make sure to keep x constant and carry it forward.0563

If you are integrating with respect to x, hold y constant, and carry it forward and do the evaluation. That is actually going to be the biggest stumbling block in this. It is not going to be the mathematics itself, it is going to be keeping track of what is going on.0571

So, example 1. Find the integral of xy3 over the region x goes from 1 to 2, and y goes from 4 to 6.0585

We have this little bit of a rectangle, we have 1, we have 2, let us say 4 is here. Let us say 6 is here, so we have this little rectangle that we are going to be evaluating this function, integrating this function over this domain.0622

Well, the integral, I will just call it i. In this particular case I have a choice since I am given the endpoints of the intervals explicitly, I can do dx dy, or I can do dy dx.0642

I am going to do dy first and save dx for last. Personal choice.0657

So I am going to integrate from 1 to 2. I am going to integrate from 4 to 6 -- we are working inside out -- xy3 dy dx.0660

okay. So, I am going to do the first integral. The inside integral first. This is going to equal the integral from 1 to 2, and it is always great to write out everything, do not keep things in your head, do not write things shorthand, write everything out.0672

It is not going to hurt if you have a couple of extra lines of mathematics, at least that way if there is a problem, you can follow it -- working your way back.0686

Now, when you integrate this, this symbol stays, we are doing this integration. We are integration with respect to y, x stays constant.0696

Well, integrate y3, the integral of y3 is y4/4, so it becomes xy4/4 evaluated from 4 to 6 and then dx... so far so good.0705

That equals the integral from 1 to 2, well when I put 6 into y and evaluate this, I am going to end up with so 64/4.0721

I get the integral of 324x -... and of course I put 4 in for y, take the fourth power, divide by 4, multiply by 6, I am going to get -64x dx.0736

That is equal to the integral from 1 to 2 of 260x dx.0752

I am going to pull the 260 out because that is just my personal preference. I like to keep the constants outside and multiply them at the end.0759

260, integral of x, dx = 260 × x2/2 evaluated from 1 to 2.0770

I just integrated this now with respect to x. I got x2/2, and when I run through this evaluation putting 2 in, subtracting putting 1 in, multiplying by 260, I end up with 390.0781

That is it. Nice and simple. Okay.0796

Let us go ahead and give a physical interpretation of what this means and it might help, it might not. In single variable calculus, we had some function and we had the interval from a to b. Well, we interpret the integral physically as the area underneath the graph.0801

Okay. We have the same thing in a function of two variables. We said in previous lessons that a function of two variables, since it is defined over a region in the x,y plane, the value of the function can actually be used as a third variable.0821

We can graph 3-space. Basically, a function of two variables is a surface in 3-space, so let us go ahead and draw a little -- so this is the x, and this is the y, and this is the z -- so there are some surface above the x,y plane.0836

Well, basically, the integral of a function of two variables over a particular region can be interpreted as the volume of everything above that region up to the surface. That is it.0861

Again, the area underneath the graph, the volume underneath the graph. We are just moving up one dimension. That is a physical interpretation that is there for you. If you want to think about it that way, that is fine.0884

I think it helps to -- you know -- in certain circumstances, but it is important to remember that an integral is a number. It is an algebraic property, not necessarily a physical property. It can be interpreted as such, but that is not what it is.0894

Let us go ahead and do another example here. So, example 2.0908

This time, f -- let us let f equal to x2y -- and r be the region such that x runs from 2 to 3 and y is the region that is above, well between, the functions x2 and x3.0918

Let us go ahead and draw this out. Again, we do not need a precise drawing... that is the x2. that is the x3, let us say this is 2 and this is 3.0962

We have that, we have that, so the region that we are looking at is that region right there. That is the area over which we are going to be integrating this particular function.0975

Okay. So, in this particular case, we are constrained by the nature of the problem to do our... to differentiate with respect to y first and then differentiate with respect to x. It is simply easier that way.0990

So, let us go ahead and do it.1005

So, the integral is equal to the integral from 2 to 3, of the integral, so the lower function is the x2, the upper in this case is the x3.1008

So, we are going to go from x2 to x3, this is perfectly acceptable.1023

The function that we are integrating is x2y dy dx, so we just need to make sure that we keep the order correct.1027

Well, this is going to equal the integral 2 to 3. Now when we integrate x2y with respect to y, we are holding x2 constant, so it is going to be y2/2... x2 y2/2.1041

We are evaluating it from x2 to x3, and then we are going to do the dx.1055

This is going to be the integral from 2 to 3 -- oops, let us see if we can eliminate as many of these stray lines as possible -- the integral from 2 to 3, so when we put x3 in for here, we get x into y because we are evaluating with respect to y.1060

This is going to be x6 × x2, so we are going to get x8/2 - ... and when we put x2 in for here, it is going to be x4 × x2, it is going to be x6/2 dx.1079

Right? So far so good. Let us go ahead and go to the next page.1099

When we do that integration, it is going to equal -- actually, let me write it again so we have it on this page -- 2 to 3 x8/2 - x6/2 dx = x9/18 - x7/14.1103

Right? 6, 7 × 2 is 14, 9 × 2 is 18. We are going to evaluate that from 2 to 3, and then when we do this evaluation we are going to end up somewhere in the neighborhood -- ahh its okay, I am just going to do an approximate... it is going to be somewhere in the neighborhood of about 918 when you actually run that.1130

The number itself, I mean it is important, but it is not that important. It is the process that is important. This is where we want to come to.1153

Now, let us go ahead and do a third example. Example 3.1163

We have f(x,y) is equal to... this time, 4xy is our function, over the region bounded by y = 0, y = x and y = -x + 3.1172

Let us go ahead and draw this region out and go ahead and see what it is we are looking at.1200

So, y = 0, that is the x axis. y = x, that is this line right here, y = -x + 3, so let us go up 3, and let us go down that way in a 45 degree angle.1207

So this is going to hit at 3 and let us go ahead and put a half-way mark here. This is actually going to be 3/2 because these have the same slope.1224

In this particular case, our region is right here, this triangle. We are going to be integrating this function over this particular triangle.1235

Of course you remember from first variable calculus, sometimes regions have to be split up simply to make the integral a little bit easier to handle.1242

In this particular case, I am going to decide to save the integration with respect to x last, the outside integral.1249

Since that is the case, I am actually going to split this into two regions. This region r, I am going to split it up into R1 and R2.1257

I am going to integrate up to here, and up to there. Then I am going to add those 2 together. The reason that I do that is the upper function from here to here is different. Here it is 0 to x, here it is 0 to -x + 3, which is why I am splitting it up.1265

Let me erase this thing. So, in this particular case, our integral over R is going to equal the integral over R1 + the integral over R2 because the integrals are additive, that is exactly right.1282

Let us go ahead and deal with the integral of -- I am going to move to the next page -- so, the integral over R1 is going to equal the integral from 0 to 3/2, that is the first half and the y value, let me draw my domain again so I have it here.1300

It is going to be that, and that, so the y value is going to go from 0 to x, and my function is 4xy dy dx.1324

That is -- excuse me -- equal to the integral from 0 to 3/2 of 2xy2, I am integrating with respect to y, I am holding that constant, y2/2, the 2 and the 4 cancel leaving me 2xy2, and I am evaluating that from 0 to x dx.1339

That is going to equal, well, 2 × the integral from 0 to 3/2... when I put x in for here, it is x2 × x is x3, so it is x3 dx.1359

This is 0, so it goes away and now that is going to equal 2 × x4/4 evaluated from 0 to 3/2.1374

When I go ahead and do that, I get 81/32, so that is the integral with respect to the first region.1389

Now we will do the integral over the second region, this region right here.1399

So, the integral over region 2 is equal to -- now we are integrating from 3/2 to 3 -- so, 3/2 to 3, and the y value is going to be 0 to -x + 3.1404

Because now the upper function is -x + 3 and again our function is 4xy dy dx.1423

Well, that is going to equal 3/2 to 3, it is going to be again 2xy2, evaluated from 0 to -x + 3.1432

When I put this in here, let us see if I have enough room, no I probably do not so let me come down here -- the integral from 0... nope, doing 3/2 to 3 now -- again, keep track of our numbers.1449

From 3/2 to 3, it is going to be 2x × -x + 32, because the 0 goes away and I am just putting this into that, dx.1464

Ohh... crazy... we do not want these crazy lines all over the place.1480

Okay. 0 dx, when I multiply this out, actually I am going ot multiply everything out here so it is fine. We will do 3/2 over 3, it is going to be 2x × x2 - 6x + 9 dx = the integral from 3/2 to 3 of 2x3 - 12 x2 + 19x dx.1490

When I go ahead and evaluate that, I use mathematical software to evaluate that, I ended up with 243 over 32, therefore the integral of f over our original region R is equal to the 81/32, the integral over the first region + 243/32, the integral of the other region = 324/32.1532

That is it. Okay. So, in this particular case, I took this region and I divided it into 2 regions and I did 2 separate integrals.1567

There is a way to actually do this as one integral by just reversing the order of integration by doing dx first and then doing the dy.1575

I will go ahead and show you what that is. I will not do the integration, but I will show you how to approach it.1584

So, let me draw the domain again. We had that, and we had this, okay? So this was 3, this was 0, we had 3/2 here, this was 3, this was y = x, or if I wrote it in terms of x, x = y.1590

Okay. This graph right here, that is the y = -x + 3. If I write it in terms of x, it is x = 3 - y.1619

Now, I am thinking about it this way. I am going to do my final integration with respect to y, which means I am going to go from 0 all the way up to 3.1630

In this particular case, there is no interference. The difference -- it says if I turn the graph this way -- the difference, this length right here, okay, that I am going to be integrating this way... this length is the difference between this function and this function.1644

In this particular case, I can go ahead and take the difference between this function and this function, and then integrate with respect to y this way, so the integral looks like this.1662

The integral from 0 to -- this is not 3, this is actually 3/2 -- 0 to, this value right here is 3/2.1681

If you solve simultaneously, this is x is 3/2, this is y is 3/2, so this y value is 3/2.1692

So 0 to 3/2, the integral... well, it is going to be... you want to keep this positive, so we are taking this function, this function, this is the lower that is the upper, so let us go 3 - y to y, 4xy, this time we are going to do dx dy.1699

That is it. So, you can do this as a single integral as long as you change your perspective and if you are integrating finally along this direction, the first integration that you do, which is going to be this way, well, this difference always stays the same in the sense that it is always going to be a difference between this function and this function.1726

That is it. The outer integral covers the entire region, I did not actually have to break this up, but the way that I chose to do it, simply because I like doing x last.1752

I had to break this region up into 2, and the reason I broke it up into 2 is because the y value is actually different depending on which function I am working with past this 3/2 mark.1764

That is it. That is our introduction to double integrals. Thank you for joining us here at educator.com, we will see you next time. Bye-bye.1778

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