- #1

- 751

- 78

## Homework Statement

Find the volume between the planes ##y=0## and ##y=x## and inside the ellipsoid ##\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1##

## The Attempt at a Solution

I understand we can approach this problem under the change of variables:

$$x=au; y= bv; z=cw$$

Thus we get:

$$V= \iiint_R \,dxdydz = abc\iiint_S \,dudvdw$$

At this point the ellipsoid has become a sphere. Thus we could use spherical coordinates to compute the volume.

My issue is with the extremes of the integral; concretely with the $\theta$ angle. I would set up the integral like this:

$$\int_{0}^{\pi / 4} d\theta \int_{0}^{\pi / 2} d\phi \int_{0}^{1} dr$$

But the stated solution is:

$$\int_{0}^{\tan^{-1} (a/b)} d\theta \int_{0}^{\pi / 2} d\phi \int_{0}^{1} dr$$

My extremes make sense to me; it is just about visualizing a sphere and two intersecting planes. But ##\tan^{-1} (a/b)## confuses me.

What's wrong and why?