Volume between cylinder and ellipsoid

[ramb]

Homework Statement

I have two surfaces, a cylinder and an ellipsoid. I want to find the volume bounded by those two surfaces. The sufaces are:
S
$$x^{2}+y^{2}=4$$

and
M
$$4x^{2}+4y^{2}+z^{2}=64$$

Homework Equations

reparameterize it to get it in the form:

$$\int\int_{D}(S-M)dA$$

If you do this and reperameterize it, you need to multiply it by the determenant of the x and y values.

The Attempt at a Solution

I reperameterized the intersection of both curves to be:

$$x=2s*cos(t)$$
$$y=2s*sin(t)$$
$$z=-8(s^{2}-4)$$

so using that integral form S - M
I got:

$$\int\int_{D}-3x^{2}-3y^{2}-z^{2}+60 dA$$

plugging in the reperamiterization I got:

=$$\int\int_{R}-3(2scos(t))^{2}-3(2ssin(t))^{2}+16s^{2}-4|X_{s}*Y_{t}-X_{t}*Y_{s}| dA$$

=$$\int\int_{R}(4s^{2}-4)(4s) dA$$

=$$16\int_0^1\int_0^{2\pi}(s^{3}-s) dtds$$= $$32\pi$$

now I'm not sure if what I've done was correct, but can someone help me to make sure if it is, whether it be by this method or another method.

thanks.

 

[LCKurtz]

First, the "volume bounded by those two surfaces" isn't very clear. Do you mean the volume inside of both? I am guessing that is what you want. If so you are making the problem way too complicated. Draw a picture. The xy domain is the interior of a circle. Use cylindrical coordinates.