91Ó°ÊÓ

Find the centroid of the plane region bounded by the given curves. Assume that the density is \(\delta \equiv 1\) for each region. $$x=0, x=4, y=0, y=6$$

Use double integration in polar coordinates to find the yolume of the solid that lies below the given surface and above the plane region \(R\) bounded by the given curve. $$z=\sqrt{x^{2}+y^{2}}: \quad r=2$$

By triple integration in cylindrical coordinates. Assume throughout that each solid has unit density unless another density function is specified. Show that the centroid of a homogeneous solid right circular cone lies on its axis threc-quarters of the way from its vertex to its base.

Find the volume of the elliptical cone bounded by \(z=\) \(\sqrt{x^{2}+4 y^{2}}\) and the plane \(z=1\). (Suggestion: Integrate first with respect to \(x\).)

Suppose that a square hole with sides of length 2 is cut symmetrically through the center of a sphere of radius \(2 .\) Show that the volume removed is given by $$ V=\int_{0}^{1} F(x) d x $$ where $$ F(x)=4 \sqrt{3-x^{2}}+4\left(4-x^{2}\right) \arcsin \frac{1}{\sqrt{4-x^{2}}} $$ Next, use a computer algebra system (or the INTEGRATE key on a calculator) to approximate the volume numerically. Finally use a computer algebra system to determine the exact value $$ V=\frac{2}{3}(19 \pi+2 \sqrt{2}-54 \arctan \sqrt{2}) $$ and verify that your numerical value is consistent with this exact value.