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Chapter 6: Problem 11
Use integration tables to find the integral. $$ \int \frac{2 x}{(1-3 x)^{2}} d x $$
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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. $$ y=\sin x, y=0, x=0, x=\pi $$
A nonnegative function \(f\) is called a probability density function if \(\int_{-\infty}^{\infty} f(t) d t=1 .\) The probability that \(x\) lies between \(a\) and \(b\) is given by \(P(a \leq x \leq b)=\int_{a}^{b} f(t) d t\) The expected value of \(x\) is given by \(E(x)=\int_{-\infty}^{\infty} t f(t) d t\). Show that the nonnegative function is a probability density function, (b) find \(P(0 \leq x \leq 4),\) and (c) find \(E(x)\).$$ f(t)=\left\\{\begin{array}{ll} \frac{2}{5} e^{-2 t / 5}, & t \geq 0 \\ 0, & t<0 \end{array}\right. $$
For what value of \(c\) does the integral \(\int_{0}^{\infty}\left(\frac{1}{\sqrt{x^{2}+1}}-\frac{c}{x+1}\right) d x\) converge? Evaluate the integral for this value of \(c\).
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.
Evaluate the definite integral. $$ \int_{-\pi / 2}^{\pi / 2}\left(\sin ^{2} x+1\right) d x $$
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