91Ó°ÊÓ

Find the Jacobian of the given transformation. $$x=u e^{v}, y=u e^{-v}$$

Sketch the graph of the spherical equation and give a corresponding \(x y\) -equation. $$\phi=\frac{\pi}{3}$$

In exercises 9 and \(10,\) determine the surface area of the portion of the plane indicated as a function of the area \(A\) of the base \(R\) of the solid and the angle \(\theta\) between the given plane and the \(x y\) -plane.

Set up and evaluate the indicated triple integral in the appropriate coordinate system. \(\iiint(x+z) d V,\) where \(Q\) is the region below \(x+2 y+3 z=6\) in the first octant.

Find the Jacobian of the given transformation. $$x=2 u v, y=3 u-v$$

Compute the volume of the solid bounded by the given surfaces. $$x=y^{2}, x=4, z=2+x \text { and } z=0$$

Use an appropriate coordinate system to compute the volume of the indicated solid. Below \(z=4-x^{2}-y^{2},\) between \(y=x, y=0\) and \(x=1\)

Find the mass and center of mass of the lamina with the given density. Lamina bounded by \(y=x^{4}\) and \(y=x^{2}, \rho(x, y)=4\)

Set up and evaluate the indicated triple integral in the appropriate coordinate system. \(\iiint_{Q}(y+2) d V,\) where \(Q\) is the region below \(x+z=4\) in the first octant between \(y=1\) and \(y=2\).

Set up and evaluate the indicated triple integral in the appropriate coordinate system. \(\iiint_{Q} z d V,\) where \(Q\) is the region between \(z=\sqrt{x^{2}+y^{2}}\) and \(z=\sqrt{4-x^{2}-y^{2}}\).