91Ó°ÊÓ

Explain how to find the center of mass of a thin plate with a variable density.

Explain how to find the center of mass of a three-dimensional object with a variable density.

Explain why \(d z r d r d \theta\) is the volume of a small "box" in cylindrical coordinates.

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 \(d y d x\). \(R\) is the triangular region with vertices \((0,0),(0,2),\) and (1,0).

Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. The region in the first quadrant bounded by \(x^{2}+y^{2}=16\)

Find the volume of the following solids using triple integrals. The solid bounded below by the cone \(z=\sqrt{x^{2}+y^{2}}\) and bounded above by the sphere \(x^{2}+y^{2}+z^{2}=8\)

Evaluate each double integral over the region \(R\) by converting it to an iterated integral. $$\iint_{R}\left(x^{2}+x y\right) d A ; R=\\{(x, y): 1 \leq x \leq 2,-1 \leq y \leq 1\\}$$

Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. The region bounded by \(y=\ln x,\) the \(x\) -axis, and \(x=e\)

Find the volume of the following solids using triple integrals. The solid bounded by the surfaces \(z=e^{y}\) and \(z=1\) over the rectangle \(\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq \ln 2\\}\)