91Ó°ÊÓ

Sketch the following polar rectangles. $$R=\\{(r, \theta): 0 \leq r \leq 5,0 \leq \theta \leq \pi / 2\\}$$

Write the integral \(\iiint_{D} f(r, \theta, z) d V\) as an iterated integral where \(D=\\{(r, \theta, z): G(r, \theta) \leq z \leq H(r, \theta), g(\theta) \leq r \leq h(\theta)\) \(\alpha \leq \theta \leq \beta\\}\).

Sketch the following systems on a number line and find the location of the center of mass. $$\begin{aligned} &m_{1}=8 \mathrm{kg} \text { located at } x=2 \mathrm{m} ; m_{2}=4 \mathrm{kg} \text { located at } x=-4 \mathrm{m}\\\ &m_{3}=1 \mathrm{kg} \text { located at } x=0 \mathrm{m} \end{aligned}$$

Write the integral \(\iiint_{D} f(\rho, \varphi, \theta) d V\) as an iterated integral, where \(D=\\{(\rho, \varphi, \theta): g(\varphi, \theta) \leq \rho \leq h(\varphi, \theta), a \leq \varphi \leq b\) \(\alpha \leq \theta \leq \beta\\}\)

Sketch the following polar rectangles. $$R=\\{(r, \theta): 2 \leq r \leq 3, \pi / 4 \leq \theta \leq 5 \pi / 4\\}$$

Evaluate the following integrals. A sketch of the region of integration may be useful. $$\int_{-1}^{1} \int_{-1}^{2} \int_{0}^{1} 6 x y z d y d x d z$$

Evaluate the following iterated integrals. $$\int_{0}^{3} \int_{-2}^{1}(2 x+3 y) d x d y$$

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=2 u+v, y=2 u$$

Sketch the following polar rectangles. $$R=\\{(r, \theta): 1 \leq r \leq 4,-\pi / 4 \leq \theta \leq 2 \pi / 3\\}$$

Evaluate the following iterated integrals. $$\int_{1}^{3} \int_{0}^{\pi / 2} x \sin y d y d x$$