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Chapter 6: Problem 8
Identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int x^{2} \cos x d x $$
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For the region bounded by the graphs of the equations, find: (a) the volume of the solid formed by revolving the region about the \(x\) -axis and (b) the centroid of the region. $$ y=\sin x, y=0, x=0, x=\pi $$
Use a computer algebra system to evaluate the definite integral. In your own words, describe how you would integrate \(\int \sec ^{m} x \tan ^{n} x d x\) for each condition. (a) \(m\) is positive and even. (b) \(n\) is positive and odd. (c) \(n\) is positive and even, and there are no secant factors. (d) \(m\) is positive and odd, and there are no tangent factors.
The region bounded by \((x-2)^{2}+y^{2}=1\) is revolved about the \(y\) -axis to form a torus. Find the surface area of the torus.
The magnetic potential \(P\) at a point on the axis of a circular coil is given by \(P=\frac{2 \pi N I r}{k} \int_{c}^{\infty} \frac{1}{\left(r^{2}+x^{2}\right)^{3 / 2}} d x\) where \(N, I, r, k,\) and \(c\) are constants. Find \(P\)
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([0, \infty)\) and \(\lim _{x \rightarrow \infty} f(x)=0\), then \(\int_{0}^{\infty} f(x) d x\) converges
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