91Ó°ÊÓ

Centroid Find the centroid of the region in the first octant that is bounded above by the cone \(z=\sqrt{x^{2}+y^{2}}\) , below by the plane \(z=0,\) and on the sides by the cylinder \(x^{2}+y^{2}=4\) and the planes \(x=0\) and \(y=0\)

Use a CAS double-integral evaluator to find the integrals . Then reverse the order of integration and evaluate, again with a CAS. \begin{equation}\int_{0}^{2} \int_{y^{3}}^{4 \sqrt{2 y}}\left(x^{2} y-x y^{2}\right) d x d y\end{equation}

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

Use a CAS double-integral evaluator to find the integrals . Then reverse the order of integration and evaluate, again with a CAS. \begin{equation}\int_{0}^{2} \int_{0}^{4-y^{2}} e^{x y} d x d y\end{equation}

Use a CAS double-integral evaluator to find the integrals . Then reverse the order of integration and evaluate, again with a CAS. \begin{equation}\int_{1}^{2} \int_{0}^{x^{2}} \frac{1}{x+y} d y d x\end{equation}

Centroid Find the centroid of the region that is bounded above by the surface \(z=\sqrt{r},\) on the sides by the cylinder \(r=4,\) and below by the \(x y\) -plane.

Use a CAS double-integral evaluator to find the integrals . Then reverse the order of integration and evaluate, again with a CAS. \begin{equation}\int_{1}^{2} \int_{y^{3}}^{8} \frac{1}{\sqrt{x^{2}+y^{2}}} d x d y\end{equation}

Centroid Find the centroid of the region cut from the solid ball \(r^{2}+z^{2} \leq 1\) by the half-planes \(\theta=-\pi / 3, r \geq 0,\) and \(\theta=\pi / 3, r \geq 0\)

Moment of inertia of solid cone 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 \(\delta=1 . )\)

Moment of inertia of solid sphere Find the moment of inertia of a solid sphere of radius \(a\) about a diameter. (Take \(\delta=1 . )\)