4.9 Change of Variables (continued)

Exercises (continued)

Change of Variables in a Triple Integral

Make the change of variables indicated by $s = x+y$ and $t = x-y$ in the double integral and set up an iterated integral in $st$ variables whose value is the original given double integral. Finally, evaluate the iterated integral.

The transformation turns the solid $S'$ in $xyz$-coordinates into a box $S$ in $stu$-coordinates. Apply the transformation to the boundries of the solid $S'$ to find $stu$-coordinate descriptions of the box $S$.

Compute and simplify the Jacobian $\frac{\partial(x,y,z)}{\partial(s,t,u)}$.

Use the transformation to perform a change of variables and evaluate $\iiint_{S'} f(x,y,z) \, dV$ by evaluating

Consider the change of variables

which is the transformation from spherical coordinates to rectangular coordinates. Determine the Jacobian of the transformation. How is the result connected to our earlier work with iterated integrals in spherical coordinates?