Evaluate $\lim _{n \rightarrow \infty} \int_0^1 \int_0^1 \cdots \int_0^1 \cos ^2\left\{\frac{\pi}{2 n}\left(x_1+x_2+\cdots+x_n\right)\right\} d x_1 d x_2 \cdots d x_n.$

In the following exercise, use a graphing utility to evaluate the triple iterated integral.

$$\int_0^\pi \int_0^2 \int_0^3 \sqrt{z^2+4} d z d r d \theta$$

Determine the value of $b$ so that the volume of the ellipsoid

$$x^2+\dfrac{y^2}{b^2}+\dfrac{z^2}{9}=1$$

is $16 \pi$.

Use a graphing utility to evaluate the iterated integral.

$\int_0^1 \int_y^{2 y} \sin (x+y) d x d y$

In the following exercise, use a graphing utility to evaluate the iterated integral.

$$\int_0^1 \int_y^{2 y} \sin (x+y) d x d y$$

Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables. x = u sin θ + v cos θ, y = u cos θ + v sin θ

In the following exercise, use the indicated change of variables to evaluate the double integral.

$$\begin{aligned} & \int_R \int \frac{x}{1+x^2 y^2} d A \\ & x=u \\ & y=\frac{v}{u} \end{aligned}$$

x = 1, xy = 5, x = 5, xy = 1

Programming Consider a continuous function $f(x, y)$ over the rectangular region $R$ with vertices $(a, c),(b, c),(a, d)$, and $(b, d)$ where $a<b$ and $c<d$. Partition the intervals $[a, b]$ and $[c, d]$ into $m$ and $n$ subintervals, so that the subintervals in a given direction are of equal length. Write a program for a graphing utility to compute the sum

$$\sum_{i=1}^n \sum_{j=1}^m f\left(x_i, y_j\right) \Delta x_i \Delta y_j \approx \int_a^b \int_c^d f(x, y) d A$$

where $\left(x_i, y_j\right)$ is the center of a representative rectangle in $R$.

Use a computer algebra system to evaluate the iterated integral. ∫_0^a ∫_0^a-x (x² + y²) dy dx

Use a computer algebra system to evaluate the iterated integral. ∫_0^4∫_0^y 2 / (x + 1)(y + 1) dx dy

Use a computer algebra system to evaluate the iterated integral. $\int_{0}^{2} \int_{0}^{4-x^{2}} e^{x y} d y d x$

In the following exercise, use a graphing utility to evaluate the iterated integral.

$$\int_0^a \int_0^{a-x}\left(x^2+y^2\right) d y d x$$

Use a graphing utility to evaluate the iterated integral.

$\int_0^a \int_0^{a-x}\left(x^2+y^2\right) d y d x$

Solve for a in the triple integral. ∫_0^1∫_0^3-a-y²∫_a^4-x-y² dz dx dy = 14/15