Exercise 19

In this problem, our goal is to find the volume of the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$.

1. Set up an iterated integral in rectangular coordinates whose value is the volume of the ellipsoid. Do so by using symmetry and taking 8 times the volume of the ellipsoid in the first octant where $x$, $y$, and $z$ are all nonnegative.

2. Explain why it makes sense to use the substitution $x = as$, $y = bt$, and $z = cu$ in order to make the region of integration simpler.

3. Compute the Jacobian of the transformation given in (b).

4. Execute the given change of variables and set up the corresponding new iterated integral in $s$, $t$, and $u$.

5. Explain why this new integral is better, but is still difficult to evaluate. What additional change of variables would make the resulting integral easier to evaluate?

6. Convert the integral from (d) to a new integral in spherical coordinates.

7. Finally, evaluate the iterated integral in (f) and hence determine the volume of the ellipsoid.