91Ó°ÊÓ

Evaluate the following iterated integrals. $$\int_{0}^{2} \int_{0}^{1} 4 x y d x d y$$

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

Evaluate the following iterated integrals. $$\int_{1}^{2} \int_{0}^{1}\left(3 x^{2}+4 y^{3}\right) d y d x$$

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

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

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

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

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

What coordinate system is suggested if the integrand of a triple integral involves \(x^{2}+y^{2} ?\)

Find the mass and center of mass of the thin rods with the following density functions. $$\rho(x)=1+\sin x, \text { for } 0 \leq x \leq \pi$$