Determine which value best approximates the volume of the solid between the $x y$-plane and the function over the region. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.)

$$f(x, y)=\sqrt{x^2+y^2}$$

$R$ : circle bounded by $x^2+y^2=9$ $(a)$ $50$ $(b)$ $500$ $(c)$ $-500$ $(d)$ $5$ $(e)$ $5000$

Use a graphing utility to approximate the iterated integral for the given values of $m$ and $n$.

$$\begin{aligned} &\int_1^4 \int_1^2 \sqrt{x^3+y^3} ~~d x d y \\ &m=6, n=4 \end{aligned}$$

Use a computer algebra system to approximate the iterated integral.

$$\begin{aligned} &\int_1^4 \int_1^2 \sqrt{x^3+y^3} ~~d x d y \\ &m=6, n=4 \end{aligned}$$

Use a computer algebra system to approximate the iterated integral. $\int_0^2 \int_x^2 \sqrt{16-x^3-y^3}$ $d y$ $d x$

Use a graphing utility to approximate the iterated integral for the given values of $m$ and $n$.

$$\begin{aligned} &\int_4^6 \int_0^2 y \cos \sqrt{x} ~~d x d y \\ &m=4, n=8 \end{aligned}$$

Use a computer algebra system to approximate the iterated integral.

$$\begin{aligned} &\int_4^6 \int_0^2 y \cos \sqrt{x} ~~d x d y \\ &m=4, n=8 \end{aligned}$$

Use a computer algebra system to approximate the iterated integral. $\int_0^2 \int_0^{4-x^2} e^{x y}$ $d y$ $d x$

Use a computer algebra system to show that both orders yield the same value. $\int_0^2 \int_{\sqrt{4-x^2}}^{4-x^2 / 4} \frac{x y}{x^2+y^2+1}$ $d y$ $d x$

Use a graphing utility to approximate the iterated integral for the given values of $m$ and $n$. $\int_0^2 \int_0^4 20 e^{-x^3 / 8}~~ d y d x$ $m=10, n=20$

Switch the order of integration. $\int_0^2 \int_{\sqrt{4-x^2}}^{4-x^2 / 4} \frac{x y}{x^2+y^2+1}$ $d y$ $d x$

Use a computer algebra system to approximate the iterated integral. $\int_0^2 \int_0^4 20 e^{-x^3 / 8}~~ d y d x$ $m=10, n=20$

Sketch the region of integration. $\int_0^2 \int_{\sqrt{4-x^2}}^{4-x^2 / 4} \frac{x y}{x^2+y^2+1}$ $d y$ $d x$

Evaluate the integral.

$$\int_{e^y}^y \frac{y \ln x}{x} d x, \quad y>0$$

Find the area of the surface given by $z=f(x, y)$ over the region $R$. (Hint: Some of the integrals are simpler in polar coordinates.)

$$f(x, y)=2+x^{3 / 2}$$

$R$ : rectangle with vertices $(0,0),(0,4),(3,4),(3,0)$