- #1

Beamsbox

- 61

- 0

## Homework Statement

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-x

^{2})

^{1/2}.

The final shape resembles a top, like a spinny gyro-like top. ;)

## Homework Equations

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 [0:2

^{1/2}], [2(pi)(x)(4-x

^{2})

^{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 [0:2

^{1/2}], [[(x)(4-x

^{2})

^{1/2}] - x

^{2}] dx

(4-x

^{2}= (2-x)(2+x)) so,

V = 2(pi)(the integral) from [0:2

^{1/2}], [[(x)((2-x)(2+x))

^{1/2}] - x

^{2}] dx

... basically lost either way... no reason to continue...

Anyone see the simple way that I'm not?

Thx.

B~

Last edited: