91Ó°ÊÓ

Find the volume of the solid inside both of the spheres \(\rho=2 \sqrt{2} \cos \phi\) and \(\rho=2\).

Sketch the solid whose volume is the indicated iterated integral. \(\int_{0}^{1} \int_{0}^{2} \frac{x}{2} d x d y\)

In Problems 19-26, evaluate by using polar coordinates. Sketch the region of integration first. $$ \int_{0}^{1} \int_{x}^{1} x^{2} d y d x $$

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Solid in the first octant bounded by the surface \(9 x^{2}+4 y^{2}=36\) and the plane \(9 x+4 y-6 z=0\)

For a solid sphere of radius \(a\), find each average distance. (a) From its center (b) From a diameter (c) From a point on its boundary (consider \(\rho=2 a \cos \phi\) )

Suppose \(X\) and \(Y\) have joint PDF $$ f(x, y)= \begin{cases}e^{-x-y}, & \text { if } x \geq 0, y \geq 0 \\ 0, & \text { otherwise }\end{cases} $$ Find (a) the joint PDF of \(U=X+Y\) and \(V=X\) (b) the marginal PDF of \(U\).

Center of mass of the tetrahedron bounded by the planes \(x+y+z=1, x=0, y=0\), and \(z=0\) if the density is proportional to the sum of the coordinates of the point

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Solid in the first octant bounded by the surface \(z=9-x^{2}-y^{2}\) and the coordinate planes

Sketch the solid whose volume is the indicated iterated integral. \(\int_{0}^{1} \int_{0}^{1}(2-x-y) d y d x\)

In Problems 19-26, evaluate by using polar coordinates. Sketch the region of integration first. $$ \int_{1}^{2} \int_{0}^{\sqrt{2 x-x^{2}}}\left(x^{2}+y^{2}\right)^{-1 / 2} d y d x $$