91Ó°ÊÓ

Find the mass and center of mass of the thin rods with the following density functions. $$\rho(x)=2-x^{2} / 16, \text { for } 0 \leq x \leq 4$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\). $$R=\\{(x, y): 1 \leq x \leq 2, x+1 \leq y \leq 2 x+4\\}$$

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

Identify and sketch the following sets in cylindrical coordinates. $$\\{(r, \theta, z): 0 \leq r \leq 3,0 \leq \theta \leq \pi / 3,1 \leq z \leq 4\\}$$

Evaluate the following iterated integrals. $$\int_{1}^{4} \int_{0}^{4} \sqrt{u v} d u d v$$

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. $$T: x=u \cos \pi v, y=u \sin \pi v$$

Find the volume of the solid below the paraboloid \(z=4-x^{2}-y^{2}\) and above the following regions. $$R=\\{(r, \theta): 0 \leq r \leq 2,0 \leq \theta \leq 2 \pi\\}$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\). $$R=\left\\{(x, y): 0 \leq x \leq 4, x^{\prime} \leq y \leq 8 \sqrt{x}\right\\}$$

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

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. $$T: x=v \sin \pi u, y=v \cos \pi u$$