Chapter 12: Problem 64
Explain why it is sometimes an advantage to change the order of integration.
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In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{2} \int_{3 y^{2}-6 y}^{2 y-y^{2}} 3 y d x d y $$
In Exercises 5 and 6 , sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. $$ \int_{0}^{2 \pi} \int_{0}^{\sqrt{3}} \int_{0}^{3-r^{2}} r d z d r d \theta $$
In Exercises 23-26, evaluate the improper iterated integral. $$ \int_{0}^{3} \int_{0}^{\infty} \frac{x^{2}}{1+y^{2}} d y d x $$
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.) \(x^{2}+y^{2}=a^{2}, 0 \leq x, 0 \leq y\) (a) \(\rho=k\) (b) \(\rho=k\left(x^{2}+y^{2}\right)\)
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. $$ \int_{0}^{2} \int_{\sqrt{4-x^{2}}}^{4-x^{2} / 4} \frac{x y}{x^{2}+y^{2}+1} d y d x $$
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