### Iterated Integrals

(requires JavaScript)

1. Find ${\int }_{0}^{3}f\left(x,y\right)\phantom{\rule{0.2em}{0ex}}dx$ and ${\int }_{0}^{4}f\left(x,y\right)\phantom{\rule{0.2em}{0ex}}dy$ if $f\left(x,y\right)=2x+3{x}^{2}y$.
$9+27y$ and $8x+24{x}^{2}$.
2. Calculate the iterated integral ${\int }_{1}^{3}{\int }_{0}^{1}\left(1+4xy\right)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy$.
$10$
3. Calculate the iterated integral ${\int }_{1}^{4}{\int }_{1}^{2}\left(\frac{x}{y}+\frac{y}{x}\right)\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx$.
$\frac{21}{2}\mathrm{ln}\left(2\right)$.
4. Calculate the double integral $\underset{R}{\iint }\frac{x{y}^{2}}{{x}^{2}+1}\phantom{\rule{0.2em}{0ex}}dA$ if $R=\left\{\left(x,y\right)|0\le x\le 1,\phantom{\rule{0.5em}{0ex}}-3\le y\le 3\right\}$.
$9\mathrm{ln}\left(2\right)$
5. Find the volume of the solid lying under the elliptic paraboloid

$\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}+z=1$

and above the rectangle $R=\left[-1,1\right]×\left[-2,2\right]$.

$\frac{166}{27}$
6. Find the average value of $f\left(x,y\right)={x}^{2}y$ over the rectangle with vertices $\left(-1,0\right)$, $\left(-1,5\right)$, $\left(1,5\right)$, and $\left(1,0\right)$.
$\frac{5}{6}$