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Chapter 12: Problem 28
Set up a double integral to find the volume of the solid bounded by the graphs of the equations. \(y=0, z=0, y=x, z=x, x=0, x=5\)
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In Exercises 57 and \(58,\) (a) sketch the region of integration, (b) switch the order of integration, and (c) use a computer algebra system to show that both orders yield the same value. $$ \int_{0}^{2} \int_{y^{3}}^{4 \sqrt{2 y}}\left(x^{2} y-x y^{2}\right) d x d y $$
In Exercises 5 and 6 , sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. $$ \int_{0}^{2 \pi} \int_{0}^{\sqrt{3}} \int_{0}^{3-r^{2}} r d z d r d \theta $$
Find \(I_{x}, I_{y}, I_{0}, \overline{\bar{x}},\) and \(\overline{\bar{y}}\) for the lamina bounded by the graphs of the equations. Use a computer algebra system to evaluate the double integrals. $$ y=0, y=b, x=0, x=a, \rho=k y $$
In Exercises \(37-42,\) sketch the region \(R\) of integration and switch the order of integration. $$ \int_{-\pi / 2}^{\pi / 2} \int_{0}^{\cos x} f(x, y) d y d x $$
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.) $$ y=\sqrt{a^{2}-x^{2}}, y=0, y=x, \rho=k \sqrt{x^{2}+y^{2}} $$
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