91Ó°ÊÓ

Find the area of the surface. The portion of the plane \(z=24-3 x-2 y\) in the first octant

Evaluate the iterated integral. $$\int_{0}^{1} \int_{0}^{x} \sqrt{1-x^{2}} d y d x$$

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.) \(y=\frac{1}{1+x^{2}}, y=0, x=-1, x=1, \rho=k\)

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

Evaluate the iterated integral by converting to polar coordinates. $$\int_{0}^{a} \int_{0}^{\sqrt{a^{2}-y^{2}}} y d x d y$$

Find the area of the surface. The portion of the paraboloid \(z=16-x^{2}-y^{2}\) in the first octant

Evaluate the iterated integral. $$\int_{-4}^{4} \int_{0}^{x^{2}} \sqrt{64-x^{3}} d y d x$$

Evaluate the iterated integral by converting to polar coordinates. $$\int_{0}^{a} \int_{0}^{\sqrt{\alpha^{2}-x^{2}}} x d y d x$$

Set up a triple integral for the volume of the solid. The solid that is the common interior below the sphere \(x^{2}+y^{2}+z^{2}=80\) and above the paraboloid \(z=\frac{1}{2}\left(x^{2}+y^{2}\right)\)

Convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. $$ \int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{0}^{\sqrt{1-x^{2}-y^{2}}} \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x $$