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Part 2 of shell method with 2 functions of y

Video transcript
In the last video, we set up this definite integral to evaluate the volume of the solid of revolution that we set up using the shell method. So now, let's just evaluate this thing. So the main thing is just simplifying this expression. I'll start off by trying to simplify this part of it. So that's going to be y plus 1. I just ate an apple, so something weird just happened in my throat. But anyway, that's done with. y plus 1 minus y squared minus 2y plus 1. I just expanded out this binomial. And then that would simplify to-- another apple in my throat moment-- so that's going to be y plus 1 minus y squared plus 2y minus 1. So this 1 and this negative 1 cancel out. And let's see. You get negative y squared plus 3y. And then we're going to multiply that times y plus 2. So when you multiply y plus 2 times this, so you have y times negative y squared, it gets us negative y to the third power. y times 3y is going to be plus 3y squared. 2 times negative y squared is negative 2y squared. And then 2 times 3y is plus 6y. So then you go all the way down here. This thing can simplify, too, because you have 3y squared minus 2y squared. So this going to be negative y to the third plus-- this part right over here simplifies to just y squared-- y squared plus 6y. So that's this entire part simplified to this down here. We can take the 2 pi out of the integral sign. So let's do that. We're integrating from y is equal to 0 to y is equal to 3 dy. And I took the 2 pi out here, and that is equal to our volume. And so now, we're essentially ready to take the antiderivative. This is going to be equal to 2 pi times the antiderivative of this business evaluated at 3 minus evaluated at 0. And I'll color code it. I found this useful. The antiderivative of y to the third is y to the fourth over 4, so this is negative y to the fourth over 4. Antiderivative of y squared is y to the third over 2-- or y to the third over 3, I should say. And then finally, I'll do it in yellow. Antiderivative of 6y is 3y squared, so plus 3y squared. And we are going to evaluate all of this business at 0 and 3. So this simplifies. This going to be equal to 2 pi times-- well, let's see. Let me do it in the same colors. 3 to the fourth power is 81. So it's negative 81 over 4, plus-- 3 to the third is 27 divided by 3 is 9, plus 9. And then 3 squared is 9 times 3 is 27, plus 27. And then when you evaluate all of these things at 0, you just get 0. So you're just subtracting out 0, so we really don't have to do anything else with the 0. And now we are ready to simplify. Let me see. Actually, let's just add them all up. So this is going to be 9 plus 27 is 36. So that is 36. And if we want to add it to negative 81 over 4, we just have to find a common denominator. So all of this business is going to be equal to 2 pi times-- and so our common denominator can be 4, times something over 4. We have negative 81 over 4, and then 36 times 4 is 144. Is that right? Yeah, that's 144. So 36 times 4, so it's plus 144. 30 times 4 is 120 plus another 24 is 144. So you have 144, essentially, minus 81. So this is going to be equal to 2 pi times-- and actually, I can even simplify it little bit more, because we have a 2 here and a 4 there. So divide the numerator and denominator by 2 so you get it over 2. So you're going to have pi times-- this is going to be 44. Let's see. If this was an 80, this would be 64. So it's going to be 63. Let me write it this way. It's going to be 63 pi over 2. Did I do that right? 60 plus 81 is 141. Add another 3, you get a 144. Yep. And we're done. We figured out the volume of our front of jet engine-looking shape.