Iterated Integrals

(requires JavaScript)

1. Find ${\displaystyle {\int }_0^{3}f<!--FUNCTION\; APPLICATION-->\left(x,y\right)dx}$ and ${\displaystyle {\int }_0^{4}f<!--FUNCTION\; APPLICATION-->\left(x,y\right)dy}$ if $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=2x+3x^{2}y$.
   $9+27y$ and $8x+24x^2$.
2. Calculate the iterated integral ${\displaystyle {\int }_1^3{\int }_0^1\left(1+4xy\right)dxdy}$.
   $10$
3. Calculate the iterated integral ${\displaystyle {\int }_1^4{\int }_1^2\left(\frac{x}{y}+\frac{y}{x}\right)dydx}$.
   $\frac{21}{2}\ln <!--FUNCTION\; APPLICATION-->\left(2\right)$.
4. Calculate the double integral ${\displaystyle \underset{R}{\iint }\frac{x{y}^2}{x^2+1}dA}$ if $R=\left\{\left(x,y\right)|0\le x\le 1,-3\le y\le 3\right\}$.
   $9\ln <!--FUNCTION\; APPLICATION-->\left(2\right)$

5. Find the volume of the solid lying under the elliptic paraboloid

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

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

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