For more information, please see full course syllabus of AP Calculus AB

For more information, please see full course syllabus of AP Calculus AB

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### Example Problems for The Definite Integral

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
- Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals
- Example II: Express the Following Limit as a Definite Integral
- Example III: Evaluate the Following Definite Integral Using the Definition
- Example IV: Evaluate the Following Integral Using the Definition
- Example V: Evaluate the Following Definite Integral by Using Areas
- Example VI: Definite Integral

- Intro 0:00
- Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals 0:11
- Example II: Express the Following Limit as a Definite Integral 5:28
- Example III: Evaluate the Following Definite Integral Using the Definition 6:28
- Example IV: Evaluate the Following Integral Using the Definition 17:06
- Example V: Evaluate the Following Definite Integral by Using Areas 25:41
- Example VI: Definite Integral 30:36

### AP Calculus AB Online Prep Course

### Transcription: Example Problems for The Definite Integral

*Hello, welcome back to www.educator.com, and welcome back to AP Calculus.*0000

*Today, we are going to do some example problems for the different integral that we introduced in the last lesson.*0004

*Let us get started.*0010

*Approximate the following definite integral using mid points and the given number of sub intervals.*0014

*This is a lot like the problems that we had before, when we are dealing with area.*0021

*We want to just go ahead and start with approximation using midpoints.*0025

*I think I will work in blue here.*0031

*We have the integral from 1 to 6.*0040

*Δx is equal to, we want 5 sub intervals.*0043

*We have 6 - 1/5 which is equal to 1.*0050

*Our Δx is equal to 1.*0056

*We are going from 1, 2, 3, 4, 5, 6.*0058

*This is 1, this is 6, we are going to be looking at these endpoints.*0069

*These are going to be our x sub i.*0074

*This is going to be x sub 1, this is going to be x sub 2, x sub 3, x sub 4, and x sub 5.*0077

*1.5, 2.5, 3.5, 4.5, and 5.5, because we are approximating with midpoints.*0086

*Let us go ahead and set up a little table here.*0099

*Let me do it over here, actually.*0104

*I have got my x sub i and I have my f(x sub i).*0106

*I have got 1.5, 2.5, 3.5, 4.5, and 5.5.*0115

*When I put these numbers into this function to find out what the y value is, I get 0.753, 1.283, 1.295, 1.012, and 0.6799.*0124

*Therefore, the integral from 1 to 6 of x³ e ⁻x dx is going to be approximately equal to the sum as i goes from 1 to 5.*0152

*5 points of the f(x sub i) Δx.*0174

*That is just equal to Δx which is 1 × the sum,*0181

*I’m sorry, I will write all of this out.*0193

*This thing is nothing more than Δx × f(x sub 1) f(x sub 2) + f(x sub 3) + f(x sub 4) + f(x sub 5).*0196

*Let us go ahead and be as explicit as possible.*0218

*That is going to equal 1 × 0.753 + 1.283 + 1.295 + 1.012 + 0.6799.*0220

*When I do all of that, I end up with 5.0229, that is my approximate integral using midpoints.*0238

*We could have used right endpoints, we could have used left endpoints, 1, 2, 3, 4, 5.*0253

*You are going to get numbers that are sort of similar.*0260

*We decided to use midpoints for this particular problem.*0262

*Just to show you what it actually looks like.*0269

*Here is my list of x values, 1.5, 2.5, 3.5, 4.5.*0272

*These are the actual values of f(x) carried out to a greater degree of precision.*0277

*Our function happens to look like this.*0283

*We were going from 1 to 6 midpoints, numbers.*0285

*Midpoints give me all the numbers.*0296

*What we found was the definite integral.*0297

*In this particular case, between 1 and 6, the function happens to be positive, it is above the x axis.*0299

*Therefore, this number that we got, the 5.02 something, it happens to be the area under the curve.*0304

*It happens to be, it is the integral from 1 to 6 of this function using midpoints.*0313

*An approximation of the integral, it just happens to correspond with the area under the curve.*0320

*Express the following limit as a definite integral.*0330

*Little practice using the actual symbol, very simple.*0333

*The limit is an goes to infinity 1 to n, here is our function.*0337

*This is going to be our integrand, this we are going to turn into dx instead of Δx.*0343

*This is going to be our lower limit integration, this is going to be our upper limit of integration.*0350

*We write this as the integral from π/3 to π cos² x/ x² dx.*0354

*We are done, this is the definition, the limit of the sum.*0369

*The summation symbol becomes the integral symbol.*0375

*That is what is happening here.*0379

*For the real problem, evaluate the following definite integral using the definition.*0390

*This is going to be a bit of a process here.*0395

*One of the reasons why we want to come up with quick ways of doing the integral was because*0398

*we do not want to take this process that we are about to go through, for this problem in the next, every time we take an integral.*0401

*This is going to equal the limit as n goes to infinity of the sum from 1 to n of the x sub i Δx.*0409

*First, we find this, in other words we find an expression for that given the fact we have f(x) and we have -2 and 4.*0436

*First, we find that.*0443

*Second, once we find that, we form the sum.*0445

*Second, we are going to evaluate the sum.*0453

*When we evaluate the sum, you will end up with the function of n.*0467

*The function of n, there is going to be n in it.*0478

*The last thing, we evaluate the limit.*0484

*Third, evaluate the limit, in other words, take n in that function that we get.*0488

*This part, take n to infinity and see what you get.*0497

*Evaluate the limit as n goes to infinity.*0503

*That is what we are going to do, first, second, third, we will get our answer.*0506

*We have -2 to 4, that is where we are integrating in.*0513

*A is equal to -2.*0517

*We know that b is equal to 4.*0520

*We know that Δx = b - a/ n is equal to 4 - -2/ n is equal to 6/n.*0523

*That is our Δx.*0536

*That takes care of our Δx part, that is going to be the 6/n.*0541

*Let us see if we can come up with something for x sub i first, some expression for x sub i,*0545

*that we can put in to f(x), in order to get f(x sub i).*0555

*We are going to take that, multiply it by the 6/ n to get this thing.*0559

*That is all we are doing here.*0565

*We have taken care of the Δx, let us see if we can find x sub i.*0568

*Let us see if we can elucidate a pattern.*0572

*We have x sub 0, that is equal to -2.*0575

*x sub 1 is equal to -2 + Δx which is 6/n.*0580

*x sub 2 = -2 + 6/ Δ n + another Δx which is + 12/n.*0587

*x sub 3 = -2 + 12/n + another Δx which is 6/n, which is 18/n.*0598

*Notice, this number is 1 × 6, 1, 1.*0609

*This number is 2 × 6, 2 is here, 2 is here.*0618

*This number is 3 × 6, 3 is here, 3 is here.*0622

*We have our pattern, our x sub i is equal to -2 + i × 6/n.*0629

*Which I’m going to go ahead and write it as 6i/ n.*0640

*Great, now we found our x sub i.*0645

*Perfect, we found our x sub i, now we are going to stick our x sub i.*0647

*This -2 + 6i/ n into this function, to find f(x sub i).*0652

*F(x sub i) is equal to 2 + 2 × x sub i which is equal to 2 + 2 × -2 + 6i/ n,*0670

*that equals 2 - 4 + 12i/ n = -2 + 12i/ n, that is our f(x sub i).*0690

*F(x sub i) × Δx is going to be -2 + 12 sub i/ n, that is my f(x sub i).*0712

*My Δx was 6/n.*0723

*Therefore, this is going to equal -12/n + 72/ n².*0728

*This is my f(x sub i) Δx.*0739

*I’m going to evaluate the sum and I'm going to take the sum as i goes from 1 to n.*0746

*We have taken care of the first part, we are going to evaluate the sum of this.*0754

*The sum of the f(x sub i), -12/n + 72i/ n².*0761

*The summation symbol distributes over that.*0776

*I get the sum i from -1 to n -12/ n + the sum of i1/ n 72i/ n².*0780

*i is the index, i has to stay under the summation symbol.*0798

*Everything else can be taken out as a constant.*0803

*In other words, there is no i here.*0805

*Therefore, I can pull out the -12n.*0808

*Here I have to leave the i but I can pull out the 72/ n².*0811

*Therefore, this is going to equal -12/n × the sum of 1 to n of 1 + 72/ n² × the sum i = 1 to n of i.*0815

*Let us rewrite that so we have it on the page.*0843

*-12/n × the sum from 1 to n of 1 + 72/ n² × the sum i from 1 to n of i.*0848

*The sum from 1 to n of 1, we just add 1 n ×, it is just n + 72/ n².*0864

*We have a closed form expression for this, remember.*0878

*It is equal to n × n + 1/ 2.*0880

*It equals -12 + 72 divided by 236, 36 × n²/ n².*0887

*This is going to be n² + n.*0898

*72n²/ 2n², gives me 36n² + 36n/ n².*0904

*I just went ahead and divided the 2 into it , and then multiplied.*0921

*72 divided by 2 is 36, 36n² 36n/ n² - 12.*0925

*36n²/ n² is 36.*0936

*36n/ n² is 36/n which is equal to 24 + 36/n.*0941

*The sum of i = 1 to n of f(x sub i) dx is equal to 24 + 36/n.*0957

*Now we evaluated the sum, we have our function of n.*0970

*Now we take the limit as n goes to infinity.*0975

*The limit as n goes to infinity of 24 + 36/n, this one goes to 0.*0980

*As n goes to infinity, we are left with 24, that is our answer.*0991

*The integral from 1 to 6 of that function, going through the entire process.*0996

*Finding the Δx, finding the x sub i, forming f(x sub i).*1002

*Multiplying f(x sub i) × Δx, evaluating the sum, and then taking the limit, gives us a final answer of 24.*1009

*That is it, run through the process.*1017

*Tedious but reasonably straightforward, as long as you have the formulas that you need.*1022

*Evaluate the following integral using the definition, same thing.*1029

*We know that this is going to equal, let us go ahead and write the definition,*1034

*so that we know what we are dealing with here.*1039

*The definition, the integral from a to b of f(x) dx equals the limit as n goes to infinity.*1041

*I think it is a good idea to write down the definition of the equation over and over again, that way you remember it.*1050

*The limit to infinity, the sum as i go from 1 to n of f(x sub i) Δx.*1058

*We are going to run through the same process.*1067

*We are going to find Δx, we are going to find an expression for x sub i.*1069

*We are going to put x sub i into f.*1073

*We are going to form f(x sub i).*1077

*We are going to multiply by Δx.*1078

*We are going to evaluate the sum of that thing.*1080

*We are going to get a function of n and then we are going to take n into infinity to get our final answer.*1082

*That is what we do.*1086

*Let us go ahead and do a is equal to 1, b is equal to 6.*1090

*Therefore, the Δx = b - a/ n which is 6 - 1/ n which is 5/n.*1097

*That takes care of the Δx.*1113

*x sub 0 that is equal to 1, x sub 1 = 1 + Δx which is 5/n.*1117

*x sub 2 = 1 + 10/ n, x sub 3 = 1 + 15/n.*1131

*We see the pattern, x sub i = 1 + 5i/ n.*1140

*Therefore, our f(x sub i) is equal to, you put this into here, into there, f(x sub i).*1152

*Therefore, it is 1 + 5i/ n² + 4 × 1 + 5i/ n – 7.*1168

*It is going to equal 1 + 10i/ n + 25i²/ n² + 4 + 20i/ n - 7*1185

*which is going to equal -2 + 30i/ n + 25i²/ n².*1204

*This is just our f(x sub i), you need to now multiply that by our Δx which is 5/n.*1219

*Our f(x sub i) × Δx is equal to -2 + 30i/ n + 25i²/ n² × 5/n.*1228

*This is going to equal -10/n + 150i/ n² + 125i²/ n³.*1249

*We have our expression, now we evaluate the sum.*1272

*The sum from 1 to n of the f(x sub i) × Δx = the sum of 1 to n of this expression.*1277

*-10/n + 150i/ n² + 125i²/ n³.*1293

*Distribute, separate, sum as i goes from 1 to n of -10/ n + the sum i from 1 to n of 150i/ n²*1306

*+ sum i from 1 to n of 125i²/ n³.*1327

*There is no i here, pull it all out.*1339

*I pull out the 150/ n², I pull out the 125/ n³ = -10/n × the sum i goes from 1 to n of 1 + 150/ n² ×*1341

*the sum as i goes from 1 to n of i + 125/ n³ × the sum as i goes from 1 to n of i².*1359

*This is going to equal -10/n × n + 150/ n² × n × n + 1/2, because that is this.*1375

*We also have an expression for this one.*1393

*This is going to be + 125/ n³ × n × n + 1 × 2n + 1/ 6.*1395

*It is just arithmetic, that is all it is.*1410

*We get -10 + 75n² + 75n/ n² + 125/ n³*1415

*× 2n³ + 3n² + n/ 6 = -10 + 75 + 75/n + 250/6.*1435

*I multiply everything, cancel everything.*1457

*+ 375/ 6n + 125/ 6n² = 640/6 + 825/ 6n + 125/ 6n².*1462

*This is our function of n, now we evaluate the limit, we take n to infinity.*1489

*The limit as n goes to infinity of this thing, 640/6 + 825/ 6n + 125/ 6n².*1498

*As n goes to infinity, this goes to 0, this goes to 0.*1514

*We are left with 640/6 or 106.67.*1519

*One painful process, we definitely need a quicker way to do this.*1530

*We have a quicker way to do this.*1534

*We actually done it with antiderivatives, we will do some more.*1535

*Evaluate the following definite integral by using areas.*1543

*6 to 30, 1/3x – 4, this is a line.*1548

*Let us go ahead and draw this out.*1553

*1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7,*1569

*8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.*1588

*Actually, I meant this to be 20 not 30.*1596

*Therefore, I’m going to go ahead and change this to 20.*1602

*1/3x – 4, 4 up 1/3, up 1/3, that takes us to 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.*1607

*We are going to get that line.*1629

*We are integrating to 20.*1636

*We are looking for the integral of this function.*1638

*They want us to do it in terms of the areas.*1641

*Therefore, I’m going to find area number 1 and I’m going to find area 2.*1643

*I'm going to add them.*1649

*Area number 1 is going to be negative because it is below the x axis.*1650

*Area number 2 is going to be positive.*1653

*When I add them up, I’m going to get some net area that is equal to the integral.*1655

*Great, nice and simple.*1660

*Therefore, this integral, the integral from 6 to 20,*1664

*What is going on here?*1677

*I’m not integrating from 6, I got all my numbers mixed up here.*1693

*That is okay, very easy to fix, I’m sorry.*1702

*The integral here is actually not from 6 to 30, it is from 0 to 20.*1704

*I do not know where the 6 to 30 came from.*1709

*Fortunately, everything still stands, let us recap.*1713

*We have a function, we graph that function.*1717

*We are integrating from 0 to 20.*1720

*We graphed it from 0 to 20.*1722

*Yes, we are finding this area which is going to be negative.*1723

*We are going to be adding it to this area.*1727

*Everything else is just fine.*1729

*This is going to be 1/3x – 4 dx is going to be area number 1 + area number 2.*1734

*This is 12, from 0 to 12, f(x) is below the x axis.*1744

*Therefore, area 1 is going to be negative.*1761

*Area 1 is going to be negative, base × height/ 2, it is just a triangle.*1764

*Base is 12, height is 4/2.*1771

*That gives me -24.*1779

*Area 2 is going to be from 12 to 20, that is just going to be positive.*1784

*It is going to be base × height/ 2.*1790

*The base of this triangle is 8, the height is 20.*1792

*I just put it into here and I end up with a height of 2.667/2.*1801

*I get 10.668.*1808

*Therefore, I just add these two together.*1813

*a1 + a2, I get a -13.32.*1818

*The integral of the function is a negative number, the net area.*1825

*More area here than there is here.*1830

*If the integral from 7 to 14 of f(x) = 27 and if the interval from 10 to 14 = 9, find the integral from 7 to 10.*1839

*We have a nice property that takes care of this for us.*1849

*We want the integral from 7 to 10 of f(x) d(x).*1853

*That equals the integral from 7 to 14 of f(x) d(x) + the integral, as long as this and this are the same, the interval from 14 to 10.*1860

*This and this are that and that.*1876

*If these are the same, I can just add these integrals to give me this final one.*1880

*Here we are fine, the integral from 7 to 14 of f(x) dx.*1889

*Here, this is the integral from 14 to 10, what they gave us is 10 to 14, lower to upper 10 to 14.*1896

*Now it is 14 to 10, it is switched.*1905

*What I have to do is, when I switch the limits, I change the sign of the integral.*1907

*It is - this is equal to negative of the 10 to 14 of the f(x) dx.*1912

*Therefore, it is just equal to 27 - 9 = 18.*1922

*Thank you so much for joining us here at www.educator.com.*1931

*We will see you next time, bye.*1933

1 answer

Last reply by: Professor Hovasapian

Wed Apr 20, 2016 1:24 AM

Post by Acme Wang on April 19, 2016

Hi Professor, In example I, why you did not take the limit? I feel a little confused.

1 answer

Last reply by: Professor Hovasapian

Thu Dec 17, 2015 1:05 AM

Post by Gautham Padmakumar on December 12, 2015

at 12:32 why did you not carry through the i while multiplying f(xi) * delta x