Exercise 21

For each of the following iterated integrals,

• sketch and label the region of integration,

• convert the integral to the other coordinate system (if given in polar, to rectangular; if given in rectangular, to polar), and

• choose one of the two iterated integrals to evaluate exactly.

1. $\int_{\pi}^{3\pi/2} \int_{0}^{3} r^3 \, dr \, d\theta$

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

3. $\int_0^{\pi/2} \int_0^{\sin(\theta)} r \sqrt{1-r^2} \, dr \, d\theta.$

4. $\int_0^{\sqrt{2}/2} \int_y^{\sqrt{1-y^2}} \cos(x^2 + y^2) \, dx \, dy.$