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Evaluating integral for shell method example

Video transcript
Where we left off in the last video, we had set up a definite integral using the shell method for this strange solid of revolution. So now, let's just evaluate the integral. And like we've seen many times in these type of problems, we really just have to do some polynomial multiplication right over here. So x minus 3 squared, well, that's pretty straightforward. That's going to be x squared minus 6x plus 9. And we're going to multiply that times x minus 1. So let's do that first. So multiply that times x minus 1. And so negative 1 times 9 is negative 9. Negative 1 times negative 6 is positive 6, positive 6x. Negative 1 times x squared is negative x squared. Now x times 9 is 9x. x times negative 6x is negative 6x squared. And then x times x squared is x to the third. And so we get x to the third minus 7x squared plus 15x minus 9. So we just multiplied x minus 3 squared times x minus 1, and then we have to multiply that times x. So we could essentially raise the degree of each of these things. At least it's easier now take the anti-derivative. It's equal to 2 pi times the definite integral from 1 to 3 of this stuff times x. So it's going to be x to the fourth minus 7x to the third power plus 15x squared minus 9x. And then, of course, dx. And I'll make the dx in that same nice blue color. Now let's just take the anti-derivative. So this is going to be equal to 2 pi times the anti-derivative of all of this business. We're going to evaluate it at 3 and subtract it when it's evaluated at 1. So the anti-derivative of x to the fourth is x to the fifth over 5. The anti-derivative of x to the third is x to the fourth over 4, and we're going to multiply that times negative 7. So it's negative 7x to the fourth over 4. And then the anti-derivative of 15 x squared, that's going to be 15 times x to the third over 3. 15 divided by 3 is 5, so it's plus 5x to the third. And then finally, the anti-derivative of negative 9x, that's going to be negative 9x squared over 2. And you can verify. If you take the derivative of this, you get this business right over here. And so this is going to be equal to 2 pi. And so let's evaluate all of this business at 3. So when you evaluate it at 3, you have 3 to the fifth power over 5. And I believe 3 to the fifth is 243, but I'll verify. 3 to the third is 27, 3 to the fourth is equal to 81, 3 to the fifth is 243. So this is going to give us some hairy math to deal with. So it's going to be 243 over 5. 3 to the fourth power, that's 81. But then we have to multiply 81 times 7. So we're going to get 567. Is that right? 81 times 7. 7 times 1 is 7, 7 times 8 is 56, so we're going to get minus 567 over 4. This is going to be really painful to do the arithmetic part. But we'll power through it. And then we have 5x to the third is 27. 27 times 5 is what? 135? I don't want to make any mistakes. 27 times 5. 7 times 5 is 35, 2 times 5 is 10. Yep. 135. So plus 135. And then finally, we have minus 9x squared. So x squared is 9 times 9 is 81. So minus 81 over 2. So that's all of this business evaluated at 3. And from that, we're going to subtract it when it's evaluated at 1. Let's do this. So we get 1/5 minus 7/4 plus 5, and then minus 9/2. And what we are left with is just a really hairy fractions problem. So I will just hope that I don't make a careless mistake at this point. So let's try to do this. This is going to be equal to 2 pi times, and if we wanted to find a common multiple here, it looks like it would have to be 20. Least common multiple of 5 and 4 and 2 is 20. So this is going to give us 243 over 5 is the same thing. 243 times 4 is going to give us 3 times 4 is 12, 4 times 4 is 16, plus 1 is 17. 2 times 4 is 8, plus 1 is 9. So we have 972 over 20. And then we have to multiply 567 times 5. So you can see the arithmetic is the most painful part here. 7 times 5 is 35, 6 times 5 is 30, plus 3 is 33, 5 times 5 is 25, plus 3 is 28. So we have 2,835 over 4, and then 135 over 20. Well, 135 times 2 is going to be 270, and then times another 10 is 2,700. So plus 2,700 over 20. Did I do that right? Yeah, that's right. And then finally, 81 over 2. That's going to be the same thing as negative 810. Let me do that same color. Negative 810 over 20. Numerator and denominator both multiplied by 10. And then let's see. Negative 1/5, that's the same thing as negative 4/20. It's going to be positive 7/4 is the same thing as positive 35/20. And it's going to be a negative 5 is the same thing as negative 100 over 20. And then finally, it's going to be a positive. I don't want make that careless mistake. I want to make sure I get the signs right. After I distribute this negative, it's going to be a positive 9/2, which is the same thing as plus 90 over 20. Did I do all the signs right? Negative 1/5, positive for this one, so positive 7/4, negative 5, and then positive 90 over 20. And so now, I just have to do some hard core addition. So let's do it. So first, I'll take all of the positives and then I'll subtract out the negatives, just to simplify it, so I have to minimize the number of times. Well, I'll add all the positives together, and then I'll add all the negatives together. And that ought to make it one subtraction problem. So 972 plus 2,700 plus 35 plus 90. So let me just write it down. So this is 2,700 plus 972. I should probably take out a calculator at this point, but I'll just do it by hand, since I've already done so much of it by hand. Plus 972 plus 90 plus 35. So we get a 7. 7 plus 9 is 16, plus 3 is 19. Did I do that right? Yeah, 16 plus 3 is 19, and this is 17, and this is a 3. So we have 3,797 when we add in all the positive numerators. And then all the negative numerators, let's see. I'm going to add them together to see how negative. So 2,835, 810, 4, and 100. So if I add 2,835, 810, let me see, 100, and 4, this is how much negative I have to subtract from that. 5 plus 4 is 9, 3 plus 1 is 4, 8 plus 8 plus 1 is 17, and then you have a 3. So we're going to subtract 3,749 from 3,797. And so this actually works out quite well. That gets us to 48. Let me make sure I haven't made a careless mistake. 2,835, 810, 100, and 4. Those are all the things I'm subtracting. 2,700, 972, 90, and 35 are all the things I added. Yep. 3,797 minus 3,749 is going to be equal to 48. So this whole expression-- we deserve a drum roll now-- is going to be equal to 2 pi times 48 over 20. And both 48 and 20 are divisible by 4. So you get 12 over 5. My brain is turning into mush now. I'm becoming paranoid that I've been making careless mistakes. We're almost there. So it becomes 12 over 5. And so our final answer, 12 times 2 is 24. So it becomes 24 pi over 5. And we are done.