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Chapter 6: Problem 16
Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} x e^{-x / 2} d x $$
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A "semi-infinite" uniform rod occupies the nonnegative \(x\) -axis. The rod has a linear density \(\delta\) which means that a segment of length \(d x\) has a mass of \(\delta d x .\) A particle of mass \(m\) is located at the point \((-a, 0)\). The gravitational force \(F\) that the rod exerts on the mass is given by \(F=\int_{0}^{\infty} \frac{G M \delta}{(a+x)^{2}} d x\) where \(G\) is the gravitational constant. Find \(F\).
Evaluate the definite integral. $$ \int_{-\pi / 2}^{\pi / 2} \cos ^{3} x d x $$
(A) find the indefinite integral in two different ways. (B) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (C) Verify analytically that the results differ only by a constant. $$ \int \sec ^{4} 3 x \tan ^{3} 3 x d x $$
In your own words, describe how you would integrate \(\int \sin ^{m} x \cos ^{n} x d x\) for each condition. (a) \(m\) is positive and odd. (b) \(n\) is positive and odd. (c) \(m\) and \(n\) are both positive and even.
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\cos \frac{x}{2}, \quad y=\sin \frac{x}{2}, \quad x=0, \quad x=\pi / 2 $$
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