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# Washer method rotating around vertical line (not y-axis), part 2

AP.CALC:
CHA‑5 (EU)
,
CHA‑5.C (LO)
,
CHA‑5.C.4 (EK)

## Video transcript

using the I guess we could call it the washer method or the ring method we were able to come up with a 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 is maybe factor out this pie so it's 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 2i squared times or 2 times negative Y squared times 2 is going to be negative 4y squared and then negative Y squared squared is plus y to the fourth plus y to the fourth and then from that we are going to subtract this thing squared we are going to subtract we're going to subtract this business squared which is going to be 4 minus 4 square roots of y plus the square root of Y squared it's just going to be Y and though all of that and then all of that dy let me write that in that same color all of that D Y and so this is going to be equal to 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 you the highest degree term here is going to be our Y to the fourth so we have we have a y to the fourth I'll write it in that same color Y to the fourth and so the next highest degree term right here is this negative 4 y squared so then you have negative we 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 for square roots of why and actually just to make it clear when we take the antiderivative I'm going right that is 4y to the one-half power and we're going to multiply all of 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 five antiderivative of negative 4y squared is negative 4/3 y to the third power antiderivative of a negative Y is negative Y squared over two and then the antiderivative of 4y to the one-half let's see we're going to increment it's going to be Y to the three halves x two-thirds we're going to get eight thirds plus 8/3 y to the three halves and let's see yep metal works out and we're going to evaluate this in at one and at zero and one and at zero and lucky for us when you evaluate it zero this whole thing turns out to be zero so this is all going to be equal to PI times evaluating all this business at one so that's going to be 1/5 minus 4/3 then a green color minus 4/3 minus 1/2 minus 1/2 so whenever you're valued one just going to do so plus 8/3 plus a 2/3 plus 8/3 and let's see what's the least common multiple over here see 5 a 3 & a 2 looks like we're gonna have a denominator of 30 so this is we could rewrite this as equal to PI and we could put everything over a denominator of 30 1/5 1/5 is 6 over 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 that other shade of green so this is minus 40 over 30 negative 1/2 that's minus 15 over 30 minus 15 over 30 and then finally 8/3 is the same thing as 80 over 30 so that's plus 80 +80 so this simplifies to so let's see we have a let's see we have a 86 - let's see 86 - fifty finally actually make sure I'm doing the math right over here so 80 so 80 minus 40 is going to get us so if 80 minus 40 gets us 40 plus 6 is 46 minus 15 is 31 so this is equal to 31 PI over 30 I have a suspicion that I might have done 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 gonna do it one more time let's see 80 minus 40 is 40 46 46 minus 10 is 36 minus another five is 31 so yes we get 31 PI over 30 for our for our volume
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