91Ó°ÊÓ

Use a computer algebra system to find the volume of the solid bounded by the graphs of the equations. $$z=9-x^{2}-y^{2}, z=0$$

Sketch the region \(R\) of integration and switch the order of integration. $$\int_{-2}^{2} \int_{0}^{\sqrt{4-x^{2}}} f(x, y) d y d x$$

Find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) $$ y=\sqrt{4-x^{2}}, z=y, z=0 $$

Explain how to change from rectangular coordinates to polar coordinates in a double integral.

Sketch the region \(R\) of integration and switch the order of integration. $$\int_{0}^{2} \int_{0}^{4-x^{2}} f(x, y) d y d x$$

Set up the double integral required to find the moment of inertia \(I\), about the given line, of the lamina bounded by the graphs of the equations. Use a computer algebra system to evaluate the double integral. \(y=\sqrt{a^{2}-x^{2}}, y=0, \rho=k y\), line: \(y=a\)

Use a computer algebra system to find the volume of the solid bounded by the graphs of the equations. \(x^{2}=9-y, z^{2}=9-y\), first octant

When evaluating a triple integral with constant limits of integration in the cylindrical coordinate system, you are integrating over a part of what solid? What is the solid when you are in spherical coordinates?

Use a computer algebra system to find the volume of the solid bounded by the graphs of the equations. $$z=\frac{2}{1+x^{2}+y^{2}}, z=0, y=0, x=0, y=-0.5 x+1$$

Find the "volume" of the "four-dimensional sphere" \(x^{2}+y^{2}+z^{2}+w^{2}=a^{2}\) by evaluating \(16 \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} \int_{0}^{\sqrt{a^{2}-x^{2}-y^{2}}} \int_{0}^{\sqrt{a^{2}-x^{2}-y^{2}-z^{2}}} d w d z d y d x\).