91Ó°ÊÓ

In Exercises \(1-10\), evaluate the integral. $$ \int_{0}^{x}(2 x-y) d y $$

In Exercises 11-16, use the indicated change of variables to evaluate the double integral. $$ \begin{array}{l} \int_{R} \int y(x-y) d A \\ x=u+v \\ y=u \end{array} $$

Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral over the region \(R\). \(\int_{R} \int \frac{y}{x^{2}+y^{2}} d A\) \(R:\) trapezoid bounded by \(y=x, y=2 x, x=1, x=2\)

Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density or densities. (Hint: Some of the integrals are simpler in polar coordinates.) $$ y=9-x^{2}, y=0, \rho=k y^{2} $$

In Exercises 17-22, use a change of variables to find the volume of the solid region lying below the surface \(z=f(x, y)\) and above the plane region \(R\). \(f(x, y)=\sqrt{(x-y)(x+4 y)}\) \(R:\) region bounded by the parallelogram with vertices \((0,0),\) (1,1),(5,0),(4,-1)

Set up the triple integrals for finding the mass and the center of mass of the solid bounded by the graphs of the equations. $$ \begin{array}{l} x=0, x=b, y=0, y=b, z=0, z=b \\ \rho(x, y, z)=k x y \end{array} $$

Use spherical coordinates to find the center of mass of the solid of uniform density. Solid lying between two concentric hemispheres of radii \(r\) and \(R,\) where \(r

In Exercises 27 and 28, use spherical coordinates to find the moment of inertia about the \(z\) -axis of the solid of uniform density. Solid bounded by the hemisphere \(\rho=\cos \phi, \pi / 4 \leq \phi \leq \pi / 2,\) and the cone \(\phi=\pi / 4\)

Find the centroid of the solid region bounded by the graphs of the equations. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) $$ z=\frac{1}{y^{2}+1}, z=0, x=-2, x=2, y=0, y=1 $$

In your own words, describe \(r\) -simple regions and \(\theta\) -simple regions.