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 $(0,0),(3,0),(2,2)$

Use a computer algebra system to approximate the iterated integral. ∫_0^π/2∫_0^π∫_0^sin θ (2 cos Φ)ρ² dp dθ dΦ

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

Give a geometric description of the region of integration when the inside and outside limits of integration are constants.

(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

Prove that lim_(n→∞) ∫_0^1∫_0^1 x^n y^n dx dy = 0.

Convert the point from rectangular coordinates to spherical coordinates. (2, 2, 4√2)

In this exercise, use polar coordinates to describe the region shown.

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.

$$∫_0^2∫_y³^4√2y (x²y - xy²) dx dy$$

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 + 2/3y^(3/2), R = {(x, y): 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 - x}

Evaluate the triple iterated integral. ∫_0^π/2∫_0^y/2∫_0^1/y sin y dz dx dy

Use a computer algebra system to evaluate the iterated integral. ∫_0^π/2 ∫_0^1+sin θ 15θr dr dθ

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.