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Chapter 12: Problem 3
Evaluate the iterated integral. $$ \int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{\cos \phi} \rho^{2} \sin \phi d \rho d \phi d \theta $$
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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 \(37-42,\) sketch the region \(R\) of integration and switch the order of integration. $$ \int_{1}^{10} \int_{0}^{\ln y} f(x, y) d x d y $$
The surfaces of a double-lobed cam are modeled by the inequalities \(\frac{1}{4} \leq r \leq \frac{1}{2}\left(1+\cos ^{2} \theta\right)\) and \(\frac{-9}{4\left(x^{2}+y^{2}+9\right)} \leq z \leq \frac{9}{4\left(x^{2}+y^{2}+9\right)}\) where all measurements are in inches. (a) Use a computer algebra system to graph the cam. (b) Use a computer algebra system to approximate the perimeter of the polar curve \(r=\frac{1}{2}\left(1+\cos ^{2} \theta\right)\). This is the distance a roller must travel as it runs against the cam through one revolution of the cam. (c) Use a computer algebra system to find the volume of steel in the cam.
The value of the integral \(I=\int_{-\infty}^{\infty} e^{-x^{2} / 2} d x\) is (a) Use polar coordinates to evaluate the improper integral $$ \begin{aligned} I^{2} &=\left(\int_{-\infty}^{\infty} e^{-x^{2} / 2} d x\right)\left(\int_{-\infty}^{\infty} e^{-y^{2} / 2} d y\right) \\ &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-\left(x^{2}+y^{2}\right) / 2} d A \end{aligned} $$ (b) Use the result of part (a) to determine \(I\). For more information on this problem, see the article "Integrating \(e^{-x^{2}}\) Without Polar Coordinates" by William Dunham in Mathematics Teacher. To view this article, go to the website ww.matharticles.com
In Exercises \(51-54,\) evaluate the iterated integral. (Note that it is necessary to switch the order of integration.) $$ \int_{0}^{2} \int_{x}^{2} x \sqrt{1+y^{3}} d y d x $$
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