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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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.
In Exercises 1-4, evaluate the iterated integral. $$ \int_{0}^{4} \int_{0}^{\pi / 2} \int_{0}^{2} r \cos \theta d r d \theta d z $$
In Exercises \(51-54,\) evaluate the iterated integral. (Note that it is necessary to switch the order of integration.) $$ \int_{0}^{1} \int_{y}^{1} \sin x^{2} d x d y $$
In Exercises \(37-42,\) sketch the region \(R\) of integration and switch the order of integration. $$ \int_{0}^{4} \int_{0}^{y} f(x, y) d x d y $$
In Exercises \(1-10\), evaluate the integral. $$ \int_{0}^{x^{3}} y e^{-y / x} d y $$
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