Problem 11
In Exercises 11-16, use the indicated change of variables to evaluate the double integral. $$ \begin{array}{l} \int_{R} \int 4\left(x^{2}+y^{2}\right) d A \\ x=\frac{1}{2}(u+v) \\ y=\frac{1}{2}(u-v) \end{array} $$
Problem 11
In Exercises 11-16, evaluate the iterated integral by converting to polar coordinates. $$ \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-y^{2}}} y d x d y $$
Problem 11
Use cylindrical coordinates to find the volume of the solid. Solid bounded by the graphs of the sphere \(r^{2}+z^{2}=a^{2}\) and the cylinder \(r=a \cos \theta\)
Problem 12
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.) $$ x y=4, x=1, x=4, \rho=k x^{2} $$
Problem 12
Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral over the region \(R\). \(\int_{R} \int \sin x \sin y d A\) \(R\) : rectangle with vertices \((-\pi, 0),(\pi, 0),(\pi, \pi / 2),(-\pi, \pi / 2)\)
Problem 12
In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{-1}^{1} \int_{-2}^{2}\left(x^{2}-y^{2}\right) d y d x $$
Problem 12
Find the area of the surface. The portion of the cone \(z=2 \sqrt{x^{2}+y^{2}}\) inside the cylinder \(x^{2}+y^{2}=4\)
Problem 12
Evaluate the iterated integral by converting to polar coordinates. $$ \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} x d y d x $$
Problem 12
Use cylindrical coordinates to find the volume of the solid. Solid inside the sphere \(x^{2}+y^{2}+z^{2}=4\) and above the upper nappe of the cone \(z^{2}=x^{2}+y^{2}\)
Problem 13
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. \(f(x, y)=2 y+x^{2}\) \(R:\) triangle with vertices (0,0),(1,0),(1,1)
