Problem 10
Evaluate the double integral \(\int_{R} \int f(r, \theta) d A,\) and sketch the region \(R\). $$ \int_{0}^{\pi / 2} \int_{0}^{1-\cos \theta}(\sin \theta) r d r d \theta $$
Problem 10
Use cylindrical coordinates to find the volume of the solid. Solid inside \(x^{2}+y^{2}+z^{2}=16\) and outside \(z=\sqrt{x^{2}+y^{2}}\)
Problem 10
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 \(z=\frac{1}{2}\left(x^{2}+y^{2}\right)\)
Problem 10
Find the area of the surface. The portion of the paraboloid \(z=16-x^{2}-y^{2}\) in the first octant
Problem 10
In Exercises \(1-10\), evaluate the integral. $$ \int_{y}^{\pi / 2} \sin ^{3} x \cos y d x $$
Problem 10
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.) \(x^{2}+y^{2}=a^{2}, 0 \leq x, 0 \leq y\) (a) \(\rho=k\) (b) \(\rho=k\left(x^{2}+y^{2}\right)\)
Problem 10
Sketch the region \(R\) and evaluate the iterated integral \(\int_{R} \int f(x, y) d A .\) $$ \int_{0}^{1} \int_{y-1}^{0} e^{x+y} d x d y+\int_{0}^{1} \int_{0}^{1-y} e^{x+y} d x d y $$
Problem 11
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.) $$ y=\sqrt{x}, y=0, x=4, \rho=k x y $$
Problem 11
In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{1} \int_{0}^{2}(x+y) d y d x $$
Problem 11
Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral over the region \(R\). \(\int_{R} \int x y d A\) \(R:\) rectangle with vertices (0,0),(0,5),(3,5),(3,0)
