### Double Integrals Over General Regions

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1. Evaluate the iterated integral ${\int }_{0}^{1}{\int }_{0}^{{x}^{2}}\left(x+2y\right)dydx$.
$\frac{9}{20}$
2. Evaluate the double integral $\underset{D}{\iint }x\mathrm{cos}\left(y\right)dA$ if $D$ is bounded by $y=0$, $y={x}^{2}$, and $x=1$.
$\frac{1-\mathrm{cos}\left(1\right)}{2}$
3. Evaluate the double integral $\underset{D}{\iint }{y}^{3}dA$ if $D$ is a triangle with vertices $\left(0,2\right)$, $\left(1,1\right)$, and $\left(3,2\right)$.
$\frac{147}{20}$
4. Find the volume of the solid under the plane $x+2y-z=0$ and above the region bounded by $y=x$ and $y={x}^{4}$.
$\frac{7}{18}$
5. Find the volume of the solid bounded by the planes $x=0$, $y=0$, $z=0$, and $x+y+z=1$.
$\frac{1}{6}$
6. Find the volume of the solid bounded by the cylinder ${x}^{2}+{y}^{2}=1$ and the planes $y=z$, $x=0$, and $z=0$ in the first octant.
$\frac{1}{3}$
7. Sketch the region of integration and change the order of integration for

${\int }_{1}^{2}{\int }_{0}^{\mathrm{ln}\left(x\right)}f\left(x,y\right)dydx$.

${\int }_{0}^{\mathrm{ln}\left(2\right)}{\int }_{{e}^{y}}^{2}f\left(x,y\right)dxdy$.
8. Evaluate the integral

${\int }_{0}^{3}{\int }_{{y}^{2}}^{9}y\mathrm{cos}\left({x}^{2}\right)dxdy$

by reversing the order of integration.

$\frac{1}{4}\mathrm{sin}\left(81\right)$
9. Express $D$ as a union of Type 1 and/or Type 2 regions and evaluate the integral $\underset{D}{\iint }{x}^{2}dA$. The curve plotted in the figure is $y=1-{x}^{2}$.

10. The double integral over a region $D$ can be written as a sum of iterated integrals

$\underset{D}{\iint }f\left(x,y\right)dA={\int }_{0}^{1}{\int }_{0}^{2y}f\left(x,y\right)dxdy+{\int }_{1}^{3}{\int }_{0}^{3-y}f\left(x,y\right)dxdy$.

Sketch the region $D$ and express the double integral as an iterated integral with reversed order of integration.