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Multiple Choice If a function \(f\) is continuous on the interval \([a,\,\infty) \), then \( \int_{a}^{\infty }f(x)\,dx\) is called [(a) a definite,(b) an infinite, (c) an improper, (d) a proper ] integral.

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Multiple Choice If the \(\lim\limits_{b \rightarrow \infty }\int_{a}^{b} f(x)\,dx\) does not exist, the improper integral \(\int_{a}^{\infty }f(x)\,dx\) [ (a) converges, (b) diverges, (c) equals \(ab].\)

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True or False If a function \(f\) is continuous for all \(x\), then the improper integral \(\int_{-\infty }^{\,\infty }f(x)\,dx\) always converges.

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True or False If a function \(f\) is continuous and nonnegative on the interval \([a,\,\infty) \), and \( \int_{a}^{\infty } f( x) \,dx\) converges, then \( \int_{a}^{\infty }f(x)\,dx\) represents the area under the graph of \(y=f(x)\) for \(x \ge a.\)

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True or False If a function \(f\) is continuous for all \(x,\) then the improper integral \(\int_{-\infty }^{\infty }f(x)\,dx=\lim\limits_{a\,\rightarrow \,\infty }\int_{-a}^{a}f(x)\,dx\).

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To determine whether the improper integral \( \int_{a}^{b} f( x) \,dx\) converges or diverges, where \(f\) is continuous on \([a,\,b),\) but is not defined at \(b\), requires finding what limit?

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\(\int_{0}^{\infty }x^{2}\,dx\)

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\(\int_{0}^{5}x^{3}dx\)

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\(\int_{2}^{3}\dfrac{dx}{x-1}\)

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\(\int_{1}^{2}\dfrac{dx}{x-1}\)

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\(\int_{0}^{1}\dfrac{1}{x}\,dx\)

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\(\int_{-1}^{1}\dfrac{x\, dx}{x^{2}+1}\)

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\(\int_{0}^{1}\dfrac{x}{x^{2}-1}\,dx\)

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\(\int_{0}^{\infty }e^{-2x}\,dx\)

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\(\int_{1}^{\infty }\dfrac{dx}{x^{3}}\)

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\(\int_{-\infty }^{-10}\dfrac{dx}{x^{2}}\)

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\(\int_{0}^{\infty }e^{2x}\,dx\)

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\(\int_{0}^{\infty }e^{-x}\,dx\)

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\(\int_{-\infty }^{-1}\dfrac{4}{x}\,dx\)

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\(\int_{1}^{\infty }\dfrac{4}{x}\,dx\)

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\(\int_{3}^{\infty }\dfrac{dx}{(x-1)^{4}} \)

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\(\int_{-\infty }^{0}\dfrac{dx}{(x-1) ^{4}}\)

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\(\int_{-\infty }^{\infty }\dfrac{dx}{x^{2}+4}\)

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\(\int_{-\infty }^{\infty }\dfrac{dx}{x^{2}+1}\)

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\(\int_{0}^{1}\dfrac{dx}{x^{2}}\)

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\(\int_{0}^{1}\dfrac{dx}{x^{3}}\)

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\(\int_{0}^{1}\dfrac{dx}{x}\)

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\(\int_{4}^{6}\dfrac{dx}{x-4}\)

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\(\int_{0}^{4}\dfrac{dx}{\sqrt{4-x}}\)

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\(\int_{1}^{5}\dfrac{x~dx}{\sqrt{5-x}}\)

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\(\int_{-1}^{1}\dfrac{dx}{\sqrt[3]{x}}\)

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\(\int_{0}^{3}\dfrac{dx}{(x-2)^{2}}\)

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\(\int_{0}^{\infty }\cos x\,dx\)

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\(\int_{0}^{\infty }\sin (\pi x) \,dx\)

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\(\int_{-\infty }^{0}e^{x}\,dx\)

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\(\int_{-\infty }^{0}e^{-x}\,dx\)

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\(\int_{0}^{\pi /2}\dfrac{x\,dx}{\sin x^{2}}\)

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\(\int_{0}^{1}\dfrac{\ln x\,dx}{x}\)

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\(\int_{0}^{1}\dfrac{dx}{1-x^{2}}\)

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\(\int_{1}^{2}\dfrac{dx}{\sqrt{x^{2}-1}}\)

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\(\int_{0}^{1}\dfrac{x\,dx}{(1-x^{2})^{2}}\)

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\(\int_{0}^{2}\dfrac{dx}{(x-1)^{2}}\)

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\(\int_{0}^{\pi /4}\tan (2x) \,dx\)

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\(\int_{0}^{\pi /2}\csc x\,dx\)

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\(\int_{0}^{\infty }\dfrac{x\,dx}{\sqrt{x+1}}\)

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\(\int_{2}^{\infty }\dfrac{dx}{x\sqrt{x^{2}-1}}\)

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\(\int_{-\infty }^{\infty }\dfrac{dx}{x^{2}+4x+5}\)

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\(\int_{-\infty }^{\infty }\dfrac{dx}{e^{x}+e^{-x}}\)

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\(\int_{-\infty }^{2}\dfrac{dx}{\sqrt{4-x}}\)

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\(\int_{-\infty }^{1}\dfrac{x\,dx}{\sqrt{2-x}}\)

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\(\int_{2}^{4}\dfrac{2x\,dx}{\sqrt[3]{x^{2}-4}}\)

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\(\int_{0}^{\pi }\dfrac{1}{1-\cos x}\,dx\)

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\(\int_{-1}^{1}\dfrac{1}{x^{3}}\,dx \)

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\(\int_{0}^{2}\dfrac{dx}{x-1}\)

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\(\int_{0}^{2}\dfrac{dx}{(x-1)^{1/3}}\)

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\(\int_{-1}^{1}\dfrac{dx}{x^{5/3}}\)

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\(\int_{1}^{2}\dfrac{dx}{(2-x)^{3/4}}\)

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\(\int_{0}^{4}\dfrac{dx}{\sqrt{8x-x^{2}}}\)

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\(\int_{a}^{3a}\dfrac{2x\,dx}{(x^{2}-a^{2})^{3/2}}\), \(a \gt 0\)

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\(\int_{0}^{3}\dfrac{x\,dx}{(9-x^{2})^{3/2}}\)

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\(\int_{0}^{\infty }xe^{-x^{2}}\,dx\)

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\(\int_{0}^{\infty }e^{-x}\sin x\,dx\)

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\(\int_{1}^{\infty }\dfrac{1}{\sqrt{x^{2}-1}}\,dx\)

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\(\int_{2}^{\infty }\dfrac{2}{\sqrt{x^{2}-4}}\,dx\)

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\(\int_{1}^{\infty }\dfrac{1+e^{-x}}{x}\,dx\)

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\(\int_{1}^{\infty }\dfrac{3e^{-x}}{x}\,dx\)

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\(\int_{1}^{\infty }\dfrac{\sin ^{2}x}{x^{2}}\,dx\)

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\(\int_{1}^{\infty }\dfrac{\cos ^{2}x}{x^{2}}\,dx\)

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\(\int_{1}^{\infty }\dfrac{dx}{(x+1) \sqrt{x}}\)

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\(\int_{1}^{\infty }\dfrac{dx}{x\sqrt{1+x^{2}}}\)

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Area Between Graphs Find the area, if it is defined, of the region enclosed by the graphs of \(y=\dfrac{1}{x+1}\) and \(y=\dfrac{1}{x+2}\) on the interval \([0,\infty )\). See the figure.

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Area Between Graphs Find the area, if it is defined, under the graph of \(y=\dfrac{1}{1+x^{2}}\) to the right of \(x=0\). See the figure below.

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Volume of a Solid of Revolution Find the volume, if it is defined, of the solid of revolution generated by revolving the region bounded by the graph of \(y=e^{-x}\) and the \(x\)-axis to the right of \(x=0\) about the \(x\)-axis. See the figure below.

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Volume of a Solid of Revolution Find the volume, if it is defined, of the solid of revolution generated by revolving the region bounded by the graph of \(y=\dfrac{1}{\sqrt{x}}\) and the \(x\)-axis to the right of \(x=1\) about the \(x\)-axis. See the figure below.

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Area Between Graphs Find the area, if it is defined, between the graph of \(y=\dfrac{8a^{3}}{x^{2}+4a^{2}}\), \(a>0\), and its horizontal asymptote.

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Drug Reaction The rate of reaction \(r\) to a given dose of a drug at time \(t\) hours after administration is given by \(r(t)=t\,e^{-t^{2}}\) (measured in appropriate units).

1. Why is it reasonable to define the total reaction as the area under the graph of \(y=r(t)\) on \([0,\infty) \)?
2. Find the total reaction to the given dose of the drug.

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Present Value of Money The present value \(PV\) of a capital asset that provides a perpetual stream of revenue that flows continuously at a rate of \(R(t)\) dollars per year is given by \[ PV= \int_{0}^{\infty }R(t)e^{-rt}\,dt \]

where \(r,\) expressed as a decimal, is the annual rate of interest compounded continuously.

1. Find the present value of an asset if it provides a constant return of \(\$100\) per year and \(r=8\%\).
2. Find the present value of an asset if it provides a return of \(R(t)=1000+80t\) dollars per year and \(r=7\%\).

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Electrical Engineering In a problem in electrical theory, the integral \(\int_{0}^{\infty }Ri^{2}\,dt\) occurs, where the current \( i=Ie^{-Rt/L}\), \(t\) is time, and \(R\), \(I\), and \(L\) are positive constants. Find the integral.

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Magnetic Potential The magnetic potential \(u\) at a point on the axis of a circular coil is given by \[ u=\dfrac{2\pi NIr}{10}\! \int_{x}^{\infty }\dfrac{dy}{(r^{2}+y^{2})^{3/2}} \]

where \(N\), \(I\), \(r\), and \(x\) are constants. Find the integral.

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Electrical Engineering The field intensity \(F\) around a long (“infinite”) straight wire carrying electric current is given by the integral \[ F=\dfrac{rIm}{10}\int_{-\infty }^{\infty }\dfrac{dy}{(r^{2}+y^{2})^{3/2}} \]

where \(r,\) \(I\), and \(m\) are constants. Find the integral.

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Work The force \(F\) of gravitational attraction between two point masses \(m\) and \(M\) that are \(r\) units apart is \(F=\dfrac{GmM}{r^{2}}\) , where \(G\) is the universal gravitational constant. Find the work done in moving the mass \(m\) along a straight-line path from \(r=1\) unit to \(r=\infty \).

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For what numbers \(a\) does \(\int_{0}^{1}x^{a}\,dx\) converge?

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\(\int_{0}^{\infty }xe^{-x}\,dx\)

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\(\int_{0}^{1}x\ln x\,dx\)

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\(\int_{0}^{\infty}e^{-x}\cos x\,dx\)

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\(\int_{0}^{\infty }\tan^{-1}x\,dx \)

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Show that \(\int_{0}^{\infty }\sin x\,dx\) and \(\int_{-\infty }^{0}\sin x\,dx\) each diverge, yet \(\lim\limits_{t\,\rightarrow \,\infty }\int_{-t}^{t}\sin x\, dx=0\).

532

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Find a function \(f\) for which \(\int_{0}^{\infty }f(x)\,dx\) and \(\int_{-\infty }^{0}f(x)\,dx\) each diverge, yet \(\lim\limits_{t\,\rightarrow \,\infty }\int_{-t}^{t}f(x)\,dx=1\).

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Use the Comparison Test for Improper Integrals to show that \(\int_{0}^{\infty }\dfrac{1}{\sqrt{2+\sin x}}\,dx\) diverges.

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Use the Comparison Test for Improper Integrals to show that \( \int_{2}^{\infty }\dfrac{\ln x}{\sqrt{x^{2}-1}}\,dx\) diverges.

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If \(n\) is a positive integer, show that:

1. \(\int_{0}^{\infty }x^{n}\,e^{-x}\,dx=n\int_{0}^{\infty }x^{n-1}\,e^{-x}\,dx\)
2. \(\int_{0}^{\infty }x^{n}\,e^{-x}\,dx=n!\)

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Show that \(\int_{e}^{\infty }\dfrac{\,dx}{x ( \ln x) ^{p}}\) converges if \(p \gt 1\) and diverges if \(p \le 1\).

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Show that \(\int_{a}^{b}\dfrac{dx}{(x-a) ^{p}}\) converges if \(0 \lt p \lt 1\) and diverges if \(p \ge 1.\)

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Show that \(\int_{a}^{b}\dfrac{dx}{(b-x)^{p}}\) converges if \(0 \lt p \lt 1\) and diverges if \(p \ge 1.\)

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Refer to Problems 93 and 94. Discuss the convergence or divergence of the integrals if \(p \le 0.\) Support your explanation with an example.

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Comparison Test for Improper Integrals Show that if two functions \(f\) and \(g\) are nonnegative and continuous on the interval \([ a,\infty ) \), and if \(f( x) \geq g( x) \) for all numbers \(x > c,\) where \(c \geq a,\) then

• If \(\int_{a}^{\infty }f( x)\, dx\) converges, then \(\int_{a}^{\infty }g( x)\,dx\) also converges.
• If \(\int_{a}^{\infty }g( x)\, dx\) diverges, then \(\int_{a}^{\infty }f( x)\, dx\) also diverges.

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\(f(x) =x\)

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\(f(x) =\cos x\)

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\(f(x) =\sin x\)

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\(f(x) =e^{x} \)

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\(f(x) =e^{ax}\)

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\(f(x) =1\)

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Find the arc length of \(y=\sqrt{x-x^{2}}-\sin ^{-1}\sqrt{x}\).

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Find \(\int_{-\infty }^{a}e^{(x-e^{x})}\,dx.\)

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Find \(\int_{-\infty }^{\infty }e^{(x-e^{x})}\,dx.\)

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1. Show that the area of the region in the first quadrant bounded by the graph of \(y=e^{-x}\) and the \(x\)-axis is divided into two equal parts by the line \(x=\ln 2\).
2. If the two regions with equal areas described in (a) are rotated about the \(x\)-axis, are the resulting volumes equal? If they are unequal, which one is larger and by how much?

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Uniform Density Function Show that the function \(f\) below is a probability density function. \[ f(x)=\left\{ \begin{array}{c@{\qquad}l@{\quad}rl} 0 & \hbox{if} & x& \lt a \\ \dfrac{1}{b-a} & \hbox{if} & a & \le x \le b, a \lt b \\ 0 & \hbox{if} & x& \gt b \end{array} \right. \]

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Exponential Density Function Show that the function \(f\) is a probability density function for \(a>0\). \[ f(x)=\left\{ \begin{array}{l@{\qquad}l@{\quad}ll} \dfrac{1}{a}e^{-x/a} & \hbox{if}&x & \ge 0 \\ 0&\hbox{if}&x & \lt 0 \end{array} \right. \]

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Find the expected value \(\mu \) of the uniform density function \(f\) given in Problem 107.

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Find the expected value \(\mu \) of the exponential probability density function \(f\) defined in Problem 108.

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Find the variance \(\sigma ^{2}\) and standard deviation \( \sigma \) of the uniform density function defined in Problem 107.

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Find the variance \(\sigma ^{2}\) and standard deviation \( \sigma \) of the exponential density function defined in Problem 108.