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# Calculating integral with shell method

## Video transcript

in the last video we were able to set up this definite integral using the shell or the hollow cylinder method in order to figure out the volume of this solid of revolution and so now let's just evaluate to this thing and really the main thing we have to do here is just to multiply what we have here out so multiply this expression out so this is going to be equal to I'll take the two pi out of the integral two pi is times the integral from 0 to 1 c2 times square root of x is 2 I'll write it as 2 square roots of X but all right it's 2 X to the one-half it'll make it a little bit easier to take the antiderivative conceptually or at least in our brain so 2 times square root of X is 2x to the 1/2 2 times negative x squared is negative 2x squared and then we have negative x times the square root of x well that's X to the first times X to the 1/2 that's going to be negative x to the three-halves power and then we have negative x times negative x squared that's going to be positive x to the third power and all of that DX and so now we're ready to take the antiderivative so this is going to be equal to 2 pi times the antiderivative of all of this business evaluated at 1 and at 0 so the antiderivative of 2 times X to the 1/2 is going to be 2 it's going to be C we're going to take X to the 3 halves times 2/3 so it's going to be 4/3 X to the 3 halves and then for this term right over here it's going to be negative 2/3 X to the 3rd negative 2/3 X to the third and you can take the derivative here to verify that you actually do get this and then right over here see if we increment to this you get X to the 5 halves and so we're going to wanted to multiply by 2/5 so - let me just in another color let's say so this one right over here it's going to be minus 2/5 minus 2/5 X to the 5 halves power let me yep that works out and then finally you're going to have X to the 4th over 4 plus I'm going in color plus X to the fourth plus X to the fourth over four that's this term right over here and now we just have to evaluate it one and zero and zero luckily all of these terms end up being a zero so that that's nice and cancels out and so we're just left with we're just lowered and canceled out it just evaluates to zero so this is just 2 pi times when you evaluate all of this business at one so that's going to be 4/3 minus 2/3 minus 2/3 minus 2/5 minus 2/5 plus 1/4 plus 1/4 and the least common multiple right over here looks like 60 so we're going to want to put all of this over denominator of 60 so it's going to be 2 pi times all of this business over a denominator of 60 and 4/3 is the same thing as 80 over 60 negative 2/3 is the same thing as negative 40 over 60 negative 2/5 is the same thing as negative 24 over 60 and then 1/4 is the same thing as 15 over 60 so this is equal to and actually this will cancel over here and you just get a 30 inner denominator so in your denominator you get a 30 and up here 80 minus 40 is 40 40 minus 24 gets us to 16 16 plus 15 is 31 so we get 31 times pi over 30 for the volume of the figure right over there