91Ó°ÊÓ

Use a change of variables to evaluate the following integrals. $$\begin{aligned} &\iiint_{D} x y d V ; D \text { is bounded by the planes } y-x=0\\\ &y-x=2, z-y=0, z-y=1, z=0, \text { and } z=3 \end{aligned}$$

Sketch each region and use a double integral to find its area. The region bounded by all leaves of the rose \(r=2 \cos 3 \theta\)

Evaluate the following integrals in spherical coordinates. \(\iiint_{D} \frac{d V}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} ; D\) is the solid between the spheres of radius 1 and 2 centered at the origin.

Evaluate the following integrals as they are written. $$\int_{0}^{4} \int_{-\sqrt{16-y^{2}}}^{\sqrt{16-y^{2}}} 2 x y d x d y$$

Draw the solid whose volume is given by the following iterated integrals. Then find the volume of the solid. $$\int_{0}^{6} \int_{1}^{2} 10 d y d x$$

Sketch each region and use a double integral to find its area. The region inside both the cardioid \(r=1+\sin \theta\) and the cardioid \(r=1+\cos \theta\)

Use a change of variables to evaluate the following integrals. \(\iiint_{D} d V ; D\) is bounded by the planes \(y-2 x=0, y-2 x=1\) \(z-3 y=0, z-3 y=1, z-4 x=0,\) and \(z-4 x=3\)

\(A\) thin plate is bounded by the graphs of \(y=e^{-x}, y=-e^{-x}, x=0,\) and \(x=L .\) Find its center of mass. How does the center of mass change as \(L \rightarrow \infty ?\)

Evaluate the following integrals in spherical coordinates. $$\int_{0}^{2 \pi} \int_{0}^{\pi / 3} \int_{0}^{4 \sec \varphi} \rho^{2} \sin \varphi d \rho d \varphi d \theta$$

Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. $$\int_{0}^{4} \int_{0}^{\sqrt{16-x^{2}}} \int_{0}^{\sqrt{16-x^{2}-z^{2}}} d y d z d x \text { in the order } d x d y d z$$