91Ó°ÊÓ

\(35-36=\) Use a computer algebra system to find the exact volume of the solid. Enclosed by $$z=x^{2}+y^{2}\quad and \quad z=2 y$$

\(37-42=\) Sketch the region of integration and change the order of integration. $$\int_{0}^{1} \int_{0}^{y} f(x, y) d x d y$$

\(37-39\) Evaluate the integral by changing to spherical coordinates. $$\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{\sqrt{x^{2}+y^{2}}}^{\sqrt{2-x^{2}-y^{2}}} x y d z d y d x$$

Use a computer algebra system to find the exact value of the integral \(\iint_{R} x^{5} y^{3} e^{x y} d A,\) where \(R=[0,1] \times[0,1] .\) Then use the CAS to draw the solid whose volume is given by the integral.

Graph the solid that lies between the surfaces \(z=e^{-x^{2}} \cos \left(x^{2}+y^{2}\right)\) and \(z=2-x^{2}-y^{2}\) for \(|x| \leqslant 1\) \(|y| \leqslant 1 .\) Use a computer algebra system to approximate the volume of this solid correct to four decimal places.

\(37-39\) Evaluate the integral by changing to spherical coordinates. $$\int_{-a}^{a} \int_{-\sqrt{a^{2}-y^{2}}}^{\sqrt{a^{2}-y^{2}}} \int_{-\sqrt{a^{2}-x^{2}-y^{2}}}^{\sqrt{a^{2}-x^{2}-y^{2}}}\left(x^{2} z+y^{2} z+z^{3}\right) d z d x d y$$

\(37-42=\) Sketch the region of integration and change the order of integration. $$\int_{0}^{2} \int_{x^{2}}^{4} f(x, y) d y d x$$

Find the mass and center of mass of the solid \(E\) with the given density function \(\rho .\) $$\begin{array}{l}{E \text { is bounded by the parabolic cylinder } z=1-y^{2} \text { and the }} \\ {\text { planes } x+z=1, x=0, \text { and } z=0 ; \quad \rho(x, y, z)=4}\end{array}$$

\(37-39\) Evaluate the integral by changing to spherical coordinates. $$\int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{2-\sqrt{4-x^{2}-y^{2}}}^{2+\sqrt{4-x^{2}-y^{2}}}\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2} d z d y d x$$

Find the mass and center of mass of the solid \(E\) with the given density function \(\rho .\) $$\begin{array}{l}{E \text { is the cube given by } 0 \leqslant x \leqslant a, 0 \leqslant y \leqslant a, 0 \leqslant z \leqslant a}; \\ {\rho(x, y, z)=x^{2}+y^{2}+z^{2}}\end{array}$$