Problems

01

Order of integration

For each of the following, set up two iterated integrals, one for each order of integration. One of them is easier to evaluate. Explain why it is, then evaluate it.

• (a) ${\iint }_{\mathcal{R}}ydA$ where $\mathcal{R}$ is the region bounded by $y=x-2$ and $x=y^2$.
• (b) ${\iint }_{\mathcal{R}}{y}^2{e}^{xy}dA$ where $\mathcal{R}$ is the region bounded by $y=x$, $y=4$, $x=0$.
• (c) ${\iint }_{\mathcal{R}}{\sin }^2(x)dA$ where $\mathcal{R}$ is the region bounded by $y=\cos (x)$, $x\in [0,\pi /2]$, $y=0$, $x=0$.
• (d) ${\iint }_{\mathcal{R}}6x^{2}dA$ where $\mathcal{R}$ is the region bounded by $y=x^3$, $y=2x+4$, $x=0$.