What is the volume inside an ellipsoid between two intersecting planes?

[JD_PM]

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?

 

[BvU]

Hi,

It would appear that if $\ \ y = x \quad \Rightarrow\ \ \arctan(1)$ is a bound for the coordinates before the change. Guess what the corresponding bound is afterwards ?

 

[Ray Vickson]

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?

Assuming that $\theta$ is the polar angle, the volume element in spherical coordinates is $r^2 \sin \theta \, dr \, d\theta \, d\phi,$ so you should have $\int_0^1 r^2 \, dr,$ not $\int_0^1 dr.$

 

[Bosko]

In polar ( spherical ) coordinates ... is not simple $$ d\theta\space d\phi \space dr $$
but $$ r d\theta\space r\sin \theta d\phi \space dr $$

Let me find the picture ... aa here it is ..

 

[Bosko]

But also you can do it without calculus ( to check the result ) by careful thinking
if you compress ellipsoid
in x direction by a times
in y direction by b times
in z direction by c times you will get nice round ball

you want to calculate volume of the slice of ball between
y=0 ... x, z plane and
x/a=y/b plane => b/a=y/x

$\theta$ goes from 0 to $\arctan{ \frac{b}{a}}$

Volume of the ball with radius r=1 is $V= \frac{4\pi}{3}$
Volume of the slice is $V= \frac{4\pi}{3} \space \frac{\arctan{\frac{b}{a}} }{2\pi}$
$V= \frac{2\pi}{3} \arctan{\frac{b}{a}}$
and when you stench back the ball to the ellipsoid
$V=abc \frac{2\pi}{3} \arctan{\frac{b}{a}}$