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Chapter 12: Problem 28
Use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations. Inside the hemisphere \(z=\sqrt{16-x^{2}-y^{2}}\) and outside the cylinder \(x^{2}+y^{2}=1\)
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In Exercises 1-4, evaluate the iterated integral. $$ \int_{0}^{4} \int_{0}^{\pi / 2} \int_{0}^{2} r \cos \theta d r d \theta d z $$
The population density of a city is approximated by the model \(f(x, y)=4000 e^{-0.01\left(x^{2}+y^{2}\right)}, x^{2}+y^{2} \leq 49,\) where \(x\) and \(y\) are measured in miles. Integrate the density function over the indicated circular region to approximate the population of the city.
In Exercises 23-26, evaluate the improper iterated integral. $$ \int_{1}^{\infty} \int_{1}^{\infty} \frac{1}{x y} d x d y $$
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. $$ \int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} d y d x $$
Sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. $$ \int_{0}^{2 \pi} \int_{0}^{\pi} \int_{2}^{5} \rho^{2} \sin \phi d \rho d \phi d \theta $$
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