Funky, all the surrounding exercises are quite easy, so I assume this is too... my brain's just not catching it...
Use cylindrical shells to find the volume of the shape formed by rotating the following around the y-axis.
The (x,y) graph before rotation: use the area enclosed by the y-axis, y=x, and y=(4-x2)1/2.
The final shape resembles a top, like a spinny gyro-like top. ;)
V = (the integral) from [a:b], [2(pi)(x)(f(x)-g(x))] dx
(damn that's easier to see written out...)
The Attempt at a Solution
Okay, I'm sure I have the correct function to start with:
V = (the integral) from [0:21/2], [2(pi)(x)(4-x2)1/2 - x] dx
(This sq. root is what's screwing me up! along with the fact that you have to distribute the x into it before integrating...)
I've tried to work through it multiple ways, here's one:
Distribute the X because I don't know any product rules at the moment.
V = 2(pi)(the integral) from [0:21/2], [[(x)(4-x2)1/2] - x2] dx
(4-x2 = (2-x)(2+x)) so,
V = 2(pi)(the integral) from [0:21/2], [[(x)((2-x)(2+x))1/2] - x2] dx
... basically lost either way... no reason to continue...
Anyone see the simple way that I'm not?