91Ó°ÊÓ

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.) \(x^{2}+y^{2}=a^{2}, 0 \leq x, 0 \leq y\) (a) \(\rho=k\) (b) \(\rho=k\left(x^{2}+y^{2}\right)\)

Convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. $$ \int_{-a}^{a} \int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}} \int_{a}^{a+\sqrt{a^{2}-x^{2}-y^{2}}} x d z d y d x $$

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 \(R\). $$ \begin{aligned} &f(x, y)=(x+y) e^{x-y}\\\ &R: \text { region bounded by the square with vertices }(4,0),(6,2) \text { , }\\\ &(4,4),(2,2) \end{aligned} $$

Write a double integral that represents the surface area of \(z=f(x, y)\) over the region \(R .\) Use a computer algebra system to evaluate the double integral. $$ \begin{aligned} &f(x, y)=2 y+x^{2}\\\ &R \text { : triangle with vertices }(0,0),(1,0),(1,1) \end{aligned} $$

Use cylindrical coordinates to verify the given formula for the moment of inertia of the solid of uniform density. Right circular cylinder: \(I_{z}=\frac{3}{2} m a^{2}\) \(r=2 a \sin \theta, \quad 0 \leq z \leq h\) Use a computer algebra system to evaluate the triple integral.

Use spherical coordinates to find the mass of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) with the given density. The density at any point is proportional to the distance between the point and the origin.

Use an iterated integral to find the area of the region bounded by the graphs of the equations. $$y=x^{3 / 2}, \quad y=2 x$$

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 \(\phi=\pi / 4\)

Product Design A company produces a spherical object of radius 25 centimeters. A hole of radius 4 centimeters is drilled through the center of the object. Find (a) the volume of the object and (b) the outer surface area of the object.

Find the average value of the function over the given solid. The average value of a continuous function \(f(x, y, z)\) over a solid region \(Q\) is $$\frac{1}{V} \iint_{Q} \int f(x, y, z) d V$$ where \(V\) is the volume of the solid region \(Q\). \(f(x, y, z)=x y z\) over the cube in the first octant bounded by the coordinate planes, and the planes \(x=3, y=3\), and \(z=3\)