91Ó°ÊÓ

Find the centroid of the solid bounded above by the sphere \(\rho=a\) and below by the cone \(\phi=\pi / 4\).

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section \(8.7 .\) Evaluate the improper integrals as iterated integrals. $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{\left(x^{2}+1\right)\left(y^{2}+1\right)} d x d y$$

Find the moment of inertia of a right circular cone of base radius 1 and height 1 about an axis through the vertex parallel to the base. (Take \(\delta=1\).)

A solid is bounded below by the cone \(z=\sqrt{x^{2}+y^{2}}\) and above by the plane \(z=1 .\) Find the center of mass and the moment of inertia about the \(z\) -axis if the density is a. \(\delta(r, \theta, z)=z\) b. \(\delta(r, \theta, z)=z^{2}\)

Improper double integral Evaluate the improper integral $$\int_{0}^{1} \int_{0}^{3} \frac{x^{2}}{(y-1)^{2 / 3}} d y d x$$