For more information, please see full course syllabus of College Calculus: Level II

For more information, please see full course syllabus of College Calculus: Level II

### Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule

**Main formula:**

Trapezoidal Rule:

Midpoint Rule:

where,

Left/Right Endpoint Rules:

where for the Left Endpoint Rule,

For the Right Endpoint Rule,

**Hints and tips:**

These formulas may seem long and complicated. Instead of memorizing them, remember the geometry on which they’re based. If you can draw the pictures of the trapezoids and rectangles, you can probably reconstruct the formulas quickly.

The Midpoint Rule is more accurate than the Trapezoid Rule, even though it is simpler and requires fewer function evaluations.

By graphing the curve that you’re estimating the area under, you can often tell whether the estimates from the various formulas will be higher or lower than the true area.

Sometimes you will not have a graph of the function or an explicit formula. Instead, you will use one of these Rules to estimate integrals based on data from charts.

### Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule

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
- Trapezoidal Rule 0:13
- Graphical Representation
- How They Work
- Formula
- Why a Trapezoid?
- Lecture Example 1 5:10
- Midpoint Rule 8:23
- Why Midpoints?
- Formula
- Lecture Example 2 11:22
- Left/Right Endpoint Rule 13:54
- Left Endpoint
- Right Endpoint
- Lecture Example 3 15:32
- Additional Example 4
- Additional Example 5

### College Calculus 2 Online Course

I. Advanced Integration Techniques | ||
---|---|---|

Integration by Parts | 24:52 | |

Integration of Trigonometric Functions | 25:30 | |

Trigonometric Substitutions | 30:09 | |

Partial Fractions | 41:22 | |

Integration Tables | 20:00 | |

Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule | 22:36 | |

Simpson's Rule | 21:08 | |

Improper Integration | 44:18 | |

II. Applications of Integrals, part 2 | ||

Arclength | 23:20 | |

Surface Area of Revolution | 28:53 | |

Hydrostatic Pressure | 24:37 | |

Center of Mass | 25:39 | |

III. Parametric Functions | ||

Parametric Curves | 22:26 | |

Polar Coordinates | 30:59 | |

IV. Sequences and Series | ||

Sequences | 31:13 | |

Series | 31:46 | |

Integral Test | 23:26 | |

Comparison Test | 22:44 | |

Alternating Series | 25:26 | |

Ratio Test and Root Test | 33:27 | |

Power Series | 38:36 | |

V. Taylor and Maclaurin Series | ||

Taylor Series and Maclaurin Series | 30:18 | |

Taylor Polynomial Applications | 50:50 |

### Transcription: Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule

*We are going to do a couple more examples here.*0000

*I want to get some more practice with the midpoint rule.*0003

*We are going to estimate the integral from 1 to 2 of x × ln(x).*0006

*Now, here b = 2, a = 1.*0011

*So, δx, which is b-a/n, is 2-1/4, so that is 1/4.*0020

*The midpoint rule says the integral is approximately equal to 1/4 × f(the midpoint of these 4 intervals).*0030

*So if we take the interval from 1 to 2 and split it into 4 pieces, that is 1 and 1/4, 1 and 1/2, 1 and 3/4.*0042

*Now we want the midpoints of those 4 integrals.*0050

*The midpoint of the first one is 1 and 1/8.*0056

*The midpoint of the second one between 1 and 1/4 and 1 and 1/2 is 1 and 3/8.*0064

*The midpoint of the third one is 1 and 5/8.*0072

*The last one is 1 and 7/8.*0076

*That is 1/4.*0081

*Now the function here, we get from the integral.*0084

*That is x ln(x), so that is 1 and 1/8, we want to plug that in there.*0088

*1 and 1/8 is 1.125 × ln(1.125).*0095

*We plug each one of these values in there and I will not write them all down, but the last one here is 1 and 7/8.*0102

*Plug that in, and we get 1.875 × ln(1.875).*0110

*That is an expression now that you could plug into your calculator.*0123

*You just take these numbers and these expressions and plug them all into your calculator.*0128

*What you come up with is .634493.*0135

*So, that tells us that our integral is approximately equal to .634493.*0143

*Notice there that we never actually solve the integral as we would have using some of our earlier techniques in Calculus 2.*0154

*We just picked different points and plugged them into the function and got an approximation of the area.*0160

*So the next example I would like to do is using the right endpoint rule.*0000

*We are going to find the integral of sin(x) from 1 to 2.*0007

*It says to use n = 4, so we will split that into 4 pieces.*0013

*δx is b-a/n, so that is 2/1/4, which is 1/4.*0019

*Then, the right endpoint rule says you take the width of these rectangles, which is 1/4, that is the δx...*0028

*... times, now you plug in for the height, you plug in the right endpoint.*0037

*So we are not going to look at the left endpoint of the first interval.*0040

*We are going to look at the right endpoint of those four intervals.*0047

*So we look at f(1 and 1/4) + f(1 and 1/2) + f(1 and 3/4) +f(2).*0050

*f(2) represents the right endpoint of the last interval there.*0065

*Now the function f here is sin(x), so we will be doing sin(1.25) + sin(1.5) + sin(1.75) + sin(2).*0073

*Again, this is something you can plug into your calculator.*0093

*When I plugged it into my calculator and simplified it down,*0100

*I got .959941 as an answer.*0103

*So, that is the approximate area under sin(x) using the right endpoint rule without actually doing any actual integration.*0112

*That is the end of the lecture on approximate integration.*0126

*We covered the trapezoid rule, the midpoint rule, and the left and right endpoint rules.*0128

*Those were all brought to you by educator.com.*0138

*This is educator.com, and today we are going to discuss three methods of integration approximation, the trapezoidal rule, the midpoint rule, and the left and right end-point rules.*0000

*The idea here is that you are trying to approximate the area under a curve*0014

*The function here is f(x) and we are trying to approximate that from x=a to x=b and we are trying to find that area.*0027

*What you have done so far in your calculus class, is you just take the integral of f and then you plug in the endpoints.*0044

*The point is that there are a lot of functions that you will not be able to take the integral of directly.*0050

*So, what we are going to try to do is find approximation techniques that do not rely on us being able to take the integral.*0055

*The idea for all of these techniques is that you start out by dividing the region between a to b into n equal partitions.*0062

*Then we are going to look at the area on each one of those.*0082

*That is the first part of the formula here.*0086

*Each one of these partitions is δx y, and δx comes from b - a, that is the total width, divided by n because there are n of these segments.*0093

*Now on the trapezoidal rule, what we are going to do is label each one of these points on the x axis.*0108

*This is x _{0}, x_{1}, x_{2} all the way up to x_{n} is b.*0115

*That is where the next part of this formula comes from.*0124

*x _{0} is a, x_{1} is a + δX because it is a and then you go over δx, x_{2} is a + 2 δX all the way up to x_{n} is a + n δx.*0127

*But, of course a + n δx is a + b - a/n.*0147

*So, that is a + b -a which is b, so xn is the same as b*0156

*We have labelled these points on the x axis, and what we are trying to do is approximate the area.*0162

*What we do to approximate the area is we are going to use several different rules.*0172

*The first one is called the trapezoidal rule.*0179

*The trapezoidal rule means that you draw little trapezoids on each of these segments, and then you find the area of these trapezoids.*0183

*The area of a trapezoid is = 1/2(base _{1} × base_{2} × height).*0200

*That area of the trapezoid is reflected in this formula right here.*0218

*The 1/2 gives you that 2 right there, the height of a trapezoid, that is the height, and that is the width of one of those trapezoids, and that is δX*0225

*Then you have base _{1} + base_{2} is, I am going to show this in red, base_{1} + base_{2}, that is for the first trapezoid.*0243

*For the next trapezoid, base _{1} + base_{2}, and so on, up to the last trapezoid, base_{1} + base_{2}.*0260

*What you are doing is you are plugging each of these x _{0}, x_{1}, x_{2} into f to get these heights*0279

*But you only have 1 of the end one and 2 of each of the middle ones.*0287

*That is why you get 1 here, and two of each of the middle ones, and one of the end one.*0297

*So, that is where the formula for the trapezoidal rule comes from.*0305

*Let us try it out on an example.*0308

*Example 1 is we are going to try to estimate the integral from 1 to 2 of sin(x) dX.*0313

*Here is 1, and here is 2, and we are going to try to estimate that using n = 4.*0322

*That means we are going to divide the region from 1 to 2 into 4 pieces.*0330

*Using the formula for the trapezoid rule, we have δx/2.*0336

*Well δx is (b - a)/n, so that is 1/4.*0341

*δx/2 is 1/8, so we are going to have 1/8 times f(x _{0}) + 2f(x_{1}) + 2f(x_{2}) + 2f(x_{3}) + only 1 of f(x_{4}).*0352

*These x _{0}'s are the division points in between 1 and 2, so this x_{0} is 1, so that is sin(1) + 2, now that is 1, x_{1} × δx is 1/4.*0385

*x _{2} can go over another unit of δx, so that is 1 and 1/2, 1 and 3/4 and finally, sin(2).*0410

*We are going to take all of that, and multiply it by 1/8.*0430

*At this point it is simply a matter of plugging all of these values into a calculator.*0433

*I have a TI calculator here, and I am going to plug in 1/8 sin(1) is 0.01745 + 2sin(1.25), which is 0.0281, and so on.*0437

*You can plug the rest of the values into a calculator.*0468

*What you get at the end simplifies down to 0.951462.*0476

*So, we say that the integral from 1 to 2 of sin(x) dX is approximately = 0.951462*0486

*Next rule we are going to learn is the midpoint rule.*0503

*It is the same idea, where you have a function that you want to integrate from a to b, and you break the region up into partitions.*0508

*So, you have x _{0} = a, x_{n} = b and a bunch of partitions in between, each partition is δx y.*0522

*Except, in each partition, instead of building trapezoids, we are going to build rectangles.*0533

*We are going to build rectangles on the height of the middle of the partition.*0542

*Here, we are going to look at the middle of the partition,*0548

*We see how tall the function is at the middle of the partition and we build a rectangle that is that height.*0555

*We do that on every rectangle.*0564

*The formula we get in total is δx, that is the width of the rectangles, times the height of these rectangles,*0575

*I have labelled the midpoints of those rectangles x _{1}* and x_{2}* and x_{n}*.*0596

*Those represent these midpoints, so that is x _{1}*, there is x_{2}*, and so on.*0616

*Those are the midpoints so x _{1}* is just x_{0} + x_{1}/2, x_{2}* is just x_{2} + x_{2}/2,*0622

*And so on and those just represent the midpoints of each of these intervals.*0632

*We plug those midpoints in to find the heights of the rectangles and estimate the area.*0640

*What we are going to do for the last rule today, is we are going to use instead of the midpoints, we will use the left endpoints of each rectangle.*0650

*Instead of having to find the midpoints, the x _{1}× and the x_{2}× and so on will be the left endpoint of each interval.*0663

*We will use those to get the heights.*0677

*We will see those in the second.*0680

*First we will do an example with the midpoint rule using the same integral as before.*0682

*Sin(x), there is 1, there is 2, again we are using n = 4 so we will break it up into 4 partitions.*0686

*Except we are going to use a slightly different formula to solve it.*0698

*Again, δx = 1/4, because that is the width of each of these rectangles, but now we are going to look at the midpoints of those 4 rectangles to find the heights.*0700

*The midpoints are, well, this is 1 right here, that is 1 and 1/4, the midpoint there is 1 and 1/8.*0715

*The next midpoint is halfway between 1 and 1/4 and 1 and 1/2 and that is 1 and 3/8.*0731

*The formula that we get is δX(f(1 + 1/8) + f(1 + 3/8) + f(1 + 5/8) + f(1 + 7/8) and that is just 1/4.*0740

*Now the function here is sin(x) so we will be doing sin(1 + 1/8), that is 1.125, + sin(1 + 3/8), that is 1.375, sin(1 + 5/8) is 1.625, and sin(1 + 7/8) is 1.875.*0771

*Now it is just an expression that we can plug into our calculator.*0800

*I worked this out ahead of time, I got 0.958944*0805

*That is our best approximation for the integral using the midpoint rule.*0827

*The next formula we want to learn is the right and left endpoint rule.*0835

*We will talk about the left endpoint rule first.*0840

*It is pretty much the same as the midpoint rule.*0845

*Again, you are drawing these rectangles except instead of using the midpoint to find the height of the rectangle, you are using the left endpoint.*0849

*So that means you start out with the exact same formula,*0862

*Except that the star points that you choose to plug in to find the heights are just the left endpoints, x _{0}, x_{1}, up to x_{n-1}.*0865

*You do not go up to x _{n}.*0876

*For the right endpoint rule it is the same formula except you use the right endpoints.*0879

*The right endpoint would be x _{1}, x_{2}, up to x_{n}.*0885

*You do not have x _{0} anymore because that is the first endpoint of the left formula.*0890

*Let me draw these in different colors here.*0896

*The left endpoints give you x _{0}, x_{1}, up to x_{n-1}, so that is the left endpoint rule.*0902

*The right endpoint, I will do that one in blue, is x _{1}, x_{2}, up to x_{n-1}, and x_{n}.*0912

*You are using the right endpoints so that is the right endpoint rule.*0928

*We will do another example.*0932

*Again we are going to figure out the integral, or estimate the integral of sin(x) from 1 to 2.*0934

*We are going to use the left endpoint rule so that means the key points that we plug in for the heights are 1, 1 + 1/4, 1 + 1/2, and 1 + 3/4.*0947

*Again δx = 1/4 and our formula says δx × f(left endpoint).*0964

*So, f(1) + f(1 + 1/4) + f(1 + 1/2) + f(1 + 3/4).*0975

*We do not go to 2 because that was the right endpoint of the left interval.*0990

*The integral is approximately equal to that.*1000

*That is sin(1) + sin(1 + 1/4) + sin(1 + 1/2) + sin(1 + 3/4).*1010

*That is something that you can plug into your calculator, and when I did that ahead of time I got 0.942984.*1022

*That is our estimation of that integral using the left endpoint rule.*1038

1 answer

Last reply by: Dr. William Murray

Fri Jun 27, 2014 4:59 PM

Post by Jorge Sardinas on June 24, 2014

Dear Dr.Murray

thank for all the wonderful lectures and i just wanted to let you know that you are an amazing teacher

P.S. I also love your trigonometry lessons.

1 answer

Last reply by: Dr. William Murray

Tue Jan 7, 2014 11:29 AM

Post by Edmund Mercado on December 20, 2013

Dr. Murray:

Isn't the calculator required to be in radian mode for these solutions?

1 answer

Last reply by: Dr. William Murray

Mon Nov 12, 2012 4:47 PM

Post by Justin Malaer on November 10, 2012

Do you guys even answer questions?

1 answer

Last reply by: Dr. William Murray

Tue May 14, 2013 9:55 AM

Post by Jess Wood on October 15, 2011

In the last example of the left endpoint rule, there is a 1/2 that is multiplied throughout the added height of the left endpoints. Is that 1/2 suppose to be 1/4? If not how did the 1/2 come about?