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Chapter 12: Problem 41
Give the number of possible orders of integration when evaluating a triple integral.
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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.) $$ y=\sin \frac{\pi x}{L}, y=0, x=0, x=L, \rho=k y $$
In Exercises \(43-50\), sketch the region \(R\) whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. $$ \int_{0}^{9} \int_{\sqrt{x}}^{3} d y d x $$
In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{\pi / 2} \int_{0}^{2 \cos \theta} r d r d \theta $$
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 \(z^{2}=x^{2}+y^{2}\)
In Exercises \(31-36,\) use an iterated integral to find the area of the region bounded by the graphs of the equations. $$ 2 x-3 y=0, \quad x+y=5, \quad y=0 $$
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