Exercise 22

For each of the following questions,

• sketch the region of integration,

• change the coordinate system in which the iterated integral is written to one of the remaining two,

• evaluate the iterated integral you deem easiest to evaluate by hand.

1. $\int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{\sqrt{x^2 + y^2}}^{\sqrt{2-x^2-y^2}} xy\, dz \, dy \, dx$

2. $\int_{0}^{\pi/2} \int_{0}^{\pi} \int_{0}^{1} \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta$

3. $\int_{0}^{2\pi} \int_{0}^{1} \int_{r}^{1} r^2 \cos(\theta) \, dz \, dr \, d\theta$