Volume: shell method (optional)
Calculating integral with shell method
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 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 2 pi out of the integral. 2 pi times the integral from 0 to 1. Let's see, 2 times the square root of x is 2-- I'll write it as 2 square roots of x. But I'll write it as 2x to the 1/2. It'll make it a little bit easier to take the antiderivative conceptually, or at least in our brain. So two times the square root of x is 2x to the 1/2. 2 times negative x squared is negative 2 x 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 3/2 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-- let's see. We're going to take x to the 3/2 times 2/3. So it's going to be 4/3 x to the 3/2. And then for this term right over here it's going to be negative 2/3 x to the third. And you could take the derivative here to verify that you actually do get this. And then right over here, let's see, if we incremented this, you get x to the 5/2. And so we're going to want to multiply by 2/5. So minus-- let me do this in another color. Let's see, so this one right over here, it's going to be minus 2/5 x to the 5/2 power. Yep, that works out. And then finally you're going to have x to the fourth over 4 plus-- let me do that in a different color-- plus x to the fourth over 4. That's this term right over here. And now we just have to evaluate at 1 and 0. And 0, luckily, all of these terms end up being a 0. So that's nice and cancels out. And so we are just left with-- we're just-- [INAUDIBLE] cancel out. It just evaluates to 0. So this is just 2 pi times when you evaluate all this business at 1. So that's going to be 4/3 minus 2/3 minus 2/5 plus 1/4. And the least common multiple right over here looks like 60, so we're going to want to put all this over a 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 same thing as 80/60. Negative 2/3 is the same thing as negative 40/60. Negative 2/5 is the same thing as negative 24/60. And then 1/4 is the same thing as 15/60. So this is equal to-- and actually this will cancel over here, and you'll just get a 30 in your 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.