Area and Volume Revisited — Question 9

Question 9

Find the volume of the ellipsoidal shell $$1\le \frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}\le 4.$$ Use the change of variables $$x=2u,\qquad y=3v,\qquad z=4w.$$

Tasks

1. Describe the transformed shell.

2. Compute the three-dimensional Jacobian and the volume.

3. Check the scaling factor geometrically.

Question 9 – Solution

Strategy. The coordinate scaling turns the ellipsoids into concentric spheres and multiplies every volume by a constant Jacobian.

Step 1: Transform the shell

The inequality becomes $$1\le u^2+v^2+w^2\le 4,$$ so the transformed region is the spherical shell $1\le\rho\le 2$.

Step 2: Jacobian and volume The derivative matrix is diagonal, so $$\left|\frac{\partial(x,y,z)}{\partial(u,v,w)}\right| =|(2)(3)(4)|=24.$$ Consequently, $$\begin{align*} V&=24\left[\frac{4\pi}{3}(2^3-1^3)\right] =24\left(\frac{28\pi}{3}\right) =\boxed{224\pi}. \end{align*}$$

Verification The map stretches lengths by factors $2$, $3$, and $4$ along the coordinate axes, so it must stretch volume by their product $24$, exactly the Jacobian used above.