Exercise 19

For each of the following iterated integrals,

• sketch the region of integration,

• write an equivalent iterated integral expression in the opposite order of integration,

• choose one of the two orders and evaluate the integral.

1. $\int_{x=0}^{x=1} \int_{y=x^2}^{y=x} xy \, dy \, dx$

2. $\int_{y=0}^{y=2} \int_{x=-\sqrt{4-y^2}}^{x=0} xy \, dx \, dy$

3. $\int_{x=0}^{x=1} \int_{y=x^4}^{y=x^{1/4}} x+y \, dy \, dx$

4. $\int_{y=0}^{y=2} \int_{x=y/2}^{x=2y} x+y \, dx \, dy$