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.)
$f(x, y)=4 x$ $R$ : square with vertices $(0,0),(4,0),(4,4),(0,4)$
(a) $-200$ (b) $600$ (c ) $50$ (d) $125$ (e) $1000$

Use a computer algebra system to evaluate the iterated integral. ∫_0^2 ∫_x^2 √16-x³-y³ dy dx

Write the limits for the double integral

$$\int_R \int f(x, y) d A$$

for both orders of integration. Compute the area of $R$ by letting $f(x, y)=1$ and integrating.
Triangle: vertices $(0,0),(3,0),(2,2)$

Find the mass and center of mass of the lamina for each density. R: triangle with vertices (0, 0), (a/2, a), (a, 0) (a)ρ=k (b)ρ=kxy

(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

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 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.

Show that the function is a joint density function and find the required probability. f(x, y) = {e^-x-y, x ≥ 0, y ≥ 0 {0, elsewhere P(0 ≤ x ≤ 1, x ≤ y ≤ 1

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

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

In this exercise, find the average value of the function over the given solid. The average value of a continuous function f(x, y, z) over a solid region Q is

$$\frac{1}{V} \iint_Q \int f(x, y, z) d V$$

where V is the volume of the solid region Q.

f(x, y, z)=x+y over the solid bounded by the sphere $x^2+y^2+z^2=2$