91Ó°ÊÓ

Find the mass of the solid bounded by the planes \(x+z=1\) \(x-z=-1, y=0,\) and the surface \(y=\sqrt{z} .\) The density of the solid is \(\delta(x, y, z)=2 y+5\).

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} 6 x d y d x$$

Evaluate the integrals in Exercises \(41-44\) by changing the order of integration in an appropriate way. $$\int_{0}^{2} \int_{0}^{4-x^{2}} \int_{0}^{x} \frac{\sin 2 z}{4-z} d y d z d x$$

Evaluate the integral \(\iint_{R}\left(x^{2}+y^{2}\right)^{-2} d A,\) where \(R\) is the region inside the circle \(x^{2}+y^{2}=2\) for \(x \leq-1.\)

Find the volume of the region cut from the solid sphere \(\rho \leq a\) by the half-planes \(\theta=0\) and \(\theta=\pi / 6\) in the first octant.

Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{3} \int_{\sqrt{x / 3}}^{1} e^{y^{3}} d y d x$$

Find the volume of the region enclosed by the cylinder \(x^{2}+y^{2}=4\) and the planes \(z=0\) and \(x+y+z=4\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. Find the volume of the solid that is bounded above by the cylinder \(z=x^{2}\) and below by the region enclosed by the parabola \(y=2-x^{2}\) and the line \(y=x\) in the \(x y\) -plane.

Find the volume of the region bounded above by the paraboloid \(z=9-x^{2}-y^{2},\) below by the \(x y\) -plane, and lying outside the cylinder \(x^{2}+y^{2}=1\).

Find the average value of the function \(f(r, \theta, z)=r\) over the solid ball bounded by the sphere \(r^{2}+z^{2}=1 .\) (This is the sphere \(\left.x^{2}+y^{2}+z^{2}=1 .\right)\)