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Evaluating the integral for washer method

Video transcript
Using the-- I guess we could call it the washer method or the ring method, we were able to come up with the definite integral for the volume of this solid of revolution right over here. So this is equal to the volume. And so in this video, let's actually evaluate this integral. So the first thing that we could do is maybe factor out this pi. So this is going to be equal to pi times the definite integral from 0 to 1. And then let's square this stuff that we have right here in green. So 2 squared is going to be 4. And then we're going to have 2 times the product of both of these terms. So 2 times negative y squared times 2 is going to be negative 4 y squared. And then negative y squared squared is plus y to the fourth. And then from that, we are going to subtract this thing squared. We're going to subtract this business squared, which is going to be 4 minus 4 square roots of y plus-- well, square root of y squared is just going to be y. And all of that dy. Let me write that in that same color. And so this is going to be equal to pi times the definite integral from 0 to 1. And let's see if we can simplify this. We have a positive 4 here, but then when you distribute this negative, you're going to have a negative 4, so that cancels with that. And let's see. The highest-degree term here is going to be our y to the fourth. So we have a y to the fourth. I'll write it in that same color. And so the next-highest-degree term right here is this negative 4 y squared. So then you have negative-- let me do that same color. We have negative 4 y squared. That's that one right over there. And then we have this y. But we have to remember we have this negative out front. So it's a negative y. So this one right over here is a negative y. And then we have a negative times a negative, which is going to give us a positive 4 square roots of y. So this is going to end up being a positive 4 square roots of y. And actually, just to make it clear when we take the antiderivative, I'm going to write that as 4 y to the 1/2 power. And we're going to multiply all that stuff by dy. Now we're ready to take the antiderivative. So it's going to be equal to pi times the antiderivative of y to the fourth is y to the fifth over 5. Antiderivative of negative 4 y squared is negative 4/3 y to the third power. Antiderivative of negative y is negative y squared over 2. And then the antiderivative of 4 y to the 1/2-- let's see. We're going to increment, so it's going to be y to the 3/2 multiplied by 2/3. We're going to get 8/3 plus 8/3 y to the 3/2. And let's see. Yep. That all works out. And we're going to evaluate this at 1 and at 0. And lucky for us, when you evaluate at 0, this whole thing turns out to be 0. So this is all going to be equal to pi times evaluating all this business at 1. So that's going to be 1/5 minus 4/3-- I'll do that in a green color-- minus 4/3 minus 1/2-- so whenever you evaluate it at 1, it's just going to be-- so plus 8/3. And let's see. What's the least common multiple over here? Let's see, a 5, a 3, and a 2. It looks like we're going to have a denominator of 30. So we can rewrite this as equal to pi, and we can put everything over a denominator of 30. 1/5 is 6/30. 4/3 is 40 over 30, so this is minus 40. That's the different shade of green. Well, actually, let me make it another shade of green. So this is minus 40/30. Negative 1/2, that's minus 15/30. And then finally, 8/3 is the same thing as 80/30, so that's plus 80. So this simplifies to-- so let's see. We have 86 minus 50-- oh, actually, let me make sure I'm doing the math right over here. So 80 minus 40 is going to get us 40, plus 6 is 46, minus 15 is 31. So this is equal to 31 pi 30. I have a suspicion that I might have done something shady in this last part right over here. So this is going to be, let's see, negative 36, negative 51, plus 80. I think that seems right. I'm going to do it one more time. Let's see. 80 minus 40 is 40, 46, 46 minus 10 is 36, minus another 5 is 31. So, yes, we get 31 pi over 30 for our volume.