(a) sketch the region of integration, (b) switch the order of integration, and (c) use a computer algebra system to show that both orders yield the same value. ∫_0^2∫_√4-x²^4-x²/4 xy / x²+y²+1 dy dx

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 $16 \pi$.

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 $R$.

(a) sketch the region of integration, (b) switch the order of integration, and (c) use a computer algebra system to show that both orders yield the same value.

$$∫_0^2∫_y³^4√2y (x²y - xy²) dx dy$$

Approximation The base of a pile of sand at a cement plant is rectangular with approximate dimensions of $20$ meters by $30$ meters. If the base is placed on the $x y$-plane with one vertex at the origin, the coordinates on the surface of the pile are $(5,5,3),(15,5,6),(25,5,4),(5,15,2),(15,15,7)$, and $(25,15,3)$. Approximate the volume of sand in the pile.

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

Determine whether the statement is true or false. If it is false, explain why or give an example. The volume of the sphere x² + y² + z² = 1 is given by the integral V = 8 ∫_0^1∫_0^1 √1-x²-y² dx dy.

Explain what is meant by an iterated integral. How is it evaluated?

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

Find the mass and center of mass of the lamina for each density. R: triangle with vertices (0, 0), (0, a), (a, a) (a)ρ=k (b)ρ=ky (c)ρ=kx

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

$$\begin{aligned} & \int_R \int\left(x y+x^2\right) d A \\ & x=u \\ & y=\frac{1}{3}(u-v) \end{aligned}$$

(x,y) = (1,3), (x,y) = (4,4), (x,y) = (4,2), (x,y) = (1,1)

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

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

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