Use a computer algebra system to evaluate the iterated integral. ∫_0^2π ∫_0^1+ cos θ 6r² cos θ dr dθ

Determine whether the statement is true or false. If it is false, explain why or give an example. If f(x, y) ≤ g(x, y) for all (x, y) in R, and both f and g are continuous over R, then ∫_R∫ f(x, y) dA ≤ ∫_R∫ g(x, y) dA.

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

$$\begin{aligned} & \int_R \int 16 x y d A \\ & x=\frac{1}{4}(u+v) \\ & y=\frac{1}{2}(v-u) \end{aligned}$$

(x,y) = (0,2), (x,y) = (1,4), (x,y) = (2,2), (x,y) = (1,0)

(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.

$$∫_0^2∫_y³^4√2y (x²y - xy²) dx dy$$

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.

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

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 $16 \pi$.

Find the Jacobian $\frac{\partial(x, y)}{\partial(u, v)}$ for the indicated change of variables.

$$x = uv, y = \frac{v}{u}$$

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. ∫_0^4∫_0^y 2 / (x + 1)(y + 1) dx dy

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

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

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