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True or False  \(\int_{2}^{3} (x^{2}+x)\, dx=\int_{2}^{3} x^{2}dx+\int_{2}^{3} x\, dx\)

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True or False \(\int_{0}^{3}5e^{x^{2}}dx=\int_{0}^{3}5\, dx\cdot \int_{0}^{3}e^{x^{2}}dx\)

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True or False \(\int_{0}^{5} (x^{3}+1) dx=\int_{0}^{-3}( x^{3}+1)dx\) \(+\int_{-3}^{5} (x^{3}+1) dx\)

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If \(f\) is continuous on an interval containing the numbers \(a,\) \(b\), and \(c,\) and if \(\int_{a}^{c}f(x)\, dx=3\) and \(\int_{c}^{b}f(x)\, dx=-5,\) then \(\int_{a}^{b}f(x) dx=\)_________.

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If a function \(f\) is continuous on the closed interval \([a,b],\) then \(\bar{y}=\dfrac{1}{b-a}\int_{a}^{b}f(x)\,dx\) is the ______ ______ of \(f\) over \([a,b].\)

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True or False  If a function \(f\) is continuous on a closed interval \([a,b]\) and if \(m\) and \(M\) denote the absolute minimum value and the absolute maximum value, respectively, of \(f\) on \([a,b]\), then \[ m\leq \int_{a}^{b}f(x)\,dx\leq M. \]

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\(\int_{1}^{3} [f(x)-g(x)]\,dx \)

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\(\int_{1}^{3} [f(x)+g(x)]\,dx \)

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\(\int_{1}^{3} [5f(x)-3g(x)]\,dx \)

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\(\int_{1}^{3}[3f(x)+4g(x)]\,dx\)

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\(\int_{1}^{5}[2f(x)-3g(x)]\,dx \)

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\(\int_{1}^{5}[f(x)-g(x)]\,dx\)

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\(\int_{0}^{1}{({t^{2}-t^{3/2}})}\,dt \)

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

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

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

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

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\(\int_{e}^{8}\dfrac{1}{2x}dx\)

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\(\int_{-\pi /4}^{\pi /4}(1+2\sec x\,\tan x)\,dx \)

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\(\int_{0}^{\pi /4}(1+\sec ^{2}x) \,dx\)

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

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\(\int_{0}^{1}({\sqrt[5]{{t^{2}}}+1})\,dt \)

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\(\int_{-2}^{3}[({x-1})\,(x+3)] \,{dx} \)

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\(\int_{0}^{1}({z^{2}+1})^{2}dz\)

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\(\int_{1}^{2}{\dfrac{x^{2}-12}{x^{4}}}\,dx \)

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\(\int_{1}^{e}\dfrac{5s^{2}+s}{s^{2}}ds\)

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\(\int_{1}^{4}\dfrac{x+1}{\sqrt{x}} dx\)

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\(\int_{1}^{9}{\dfrac{\sqrt{x}+1}{x^{2}}} dx \)

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\(\int_{1}^{2}{\dfrac{2{x^{4}+1}}{x^{4}}} dx\)

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

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\(\int_{0}^{1/2}\left( 5+\dfrac{1}{\sqrt{1-x^{2}}} \right) dx\)

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\(\int_{0}^{1}\left(1+\dfrac{5}{1+x^{2}}\right) dx\)

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\(\int_{-2}^{1}\,f(x)\,dx\), where \(f(x)={{\left\{ \begin{array}{c@{ }c@{ }c} 1 & \hbox{if} & x\lt0 \\ x^{2}+1 & \hbox{if} & x\geq 0 \end{array} \right.}}\)

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\({{\int_{-1}^{-2}\,}}f(x)\,dx\), where \(f(x)={ {\left\{ \begin{array}{c@{ }c@{ }c} x+1 & \hbox{if} & x\lt0 \\ x^{2}+1 & \hbox{if} & x\geq 0 \end{array} \right.}}\)

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\(\int_{-2}^{2}\,f(x)\,dx\), where \(f(x)={{\left\{ \begin{array}{c@{ }c@{ }c} 3x & \hbox{if} & -2\leq x\lt0 \\ 2x^{2} & \hbox{if} & 0\leq x\leq 2 \end{array} \right. }}\)

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\(\int_{0}^{4}h(x)\,dx\), where \(h(x)={{\left\{ \begin{array}{c@{ }c@{ }c} x-2 & \hbox{if} & 0\leq x\leq 2 \\ 2-x & \hbox{if} & 2\lt x\leq 4 \end{array} \right. }}\)

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\(\int_{-2}^{1}H(x)\,dx\), where \(H(x)={{\left\{ \begin{array}{c@{ }c@{ }c} 1+x^{2} & \hbox{if} & -2\leq x\lt0 \\ 1+3x & \hbox{if} & 0\leq x\leq 1 \end{array} \right. }}\)

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\(\int_{-\pi /2}^{\pi /2}f(x)\,dx\), where \(f(x)=\left\{ \begin{array}{@{}l@{ }l@{ }l} {x^{2}+x} & \hbox{{if}} & {\ -{\dfrac{{\pi }}{{2}}}\leq x\leq 0} \\ {\sin x} & \hbox{{if}} & {0\lt x\lt{\dfrac{{\pi }}{{4}}}} \\ \dfrac{\sqrt{2}}{2} & \hbox{{if}} & {{\dfrac{{\pi }}{{4}}}\leq x\leq {\dfrac{{ \pi }}{{2}}}} \end{array} \right.\)

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\(\int_{3}^{11}{f(x)\,dx-{\int_{7}^{11}{f(x)\,dx}}}=\int_{3}^{7}{f(x)\,dx}{}\)

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\(\int_{-2}^{6}f(x)\,dx-{\int_{3}^{6}{f(x)\,dx}=\int_{-2}^{3}{f(x)\,dx}}\)

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\(\int_{0}^{4}{f(x)\,dx-{\int_{6}^{4}}}f(x)\,dx=\int_{0}^{6}f(x)\,dx \)

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\(\int_{-1}^{3}f(x)\,dx-{\int_{5}^{3}{f(x)\,dx=\int_{-1}^{5}{f(x)\,dx}}}\)

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\(\int_{1}^{3}{({5x+1})\,dx} \)

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

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

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\(\int_{\pi /6}^{\pi /3}{\cos x\,dx}\)

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\(\int_{0}^{1}{\sqrt{1+x^{2}}\,dx }\)

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\(\int_{-1}^{1}{\sqrt{1+x^{4}}\,dx}\)

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

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\(\int_{1}^{10}\dfrac{1}{x}dx \)

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\(\int_{0}^{3} (2{x^{2} + 1}) dx\)

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

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

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

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

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\(\int_{-\pi/4}^{\pi /4}\sec x\tan x dx\)

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\(f(x) =e^{x}\) over \([0,1]\)

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\(f(x) =\dfrac{1}{x}\) over \([1,e] \)

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\(f(x) =x^{2/3}\) over \([-1,1] \)

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\(f(x) =\sqrt{x}\) over \([0,4] \)

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\(f(x) =\sin x\) over \(\left[0,\dfrac{{\pi }}{2}\right] \)

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\(f(x) =\cos x\) over \(\left[ 0,\dfrac{{\pi }}{2} \right]\)

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\(f(x) =1-x^{2}\) over \([-1,1] \)

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\(f(x) =16-x^{2}\) over \([-4,\,4] \)

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\(f(x) =e^{x}-{\sin x}\) over \(\left[0,\dfrac{\pi}{2}\right] \)

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\(f(x) =x+\cos x\) over \(\left[0,\dfrac{{\pi }}{2}\right] \)

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\([-1, 2]\)

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\([-2, 1]\)

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\([-1, 2]\)

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\([0, \frac{3\pi}{4}]\)

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\(\int_{-2}^{3}{({x+{|x|}})\,dx}\)

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\(\int_{0}^{3}{{|x-1|}\,dx} \)

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\(\int_{0}^{2}{{|3x-1|}\,dx} \)

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

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Average Temperature  A rod \(3\) meters long is heated to \(25x^{\circ}{\rm C}\), where \(x\) is the distance in meters from one end of the rod. Find the average temperature of the rod.

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Average Daily Rainfall The rainfall per day, \(x\) days after the beginning of the year, is modeled by the function \(r(x)=0.00002(6511+366x-x^{2}) \), measured in centimeters. Find the average daily rainfall for the first \(180\) days of the year.

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Structural Engineering  A structural engineer designing a member of a structure must consider the forces that will act on that member. Most often, natural forces like snow, wind, or rain distribute force over the entire member. For practical purposes, however, an engineer determines the distributed force as a single resultant force acting at one point on the member. If the distributed force is given by the function \(W=W(x)\), in newtons per meter (N/m), then the magnitude \(F_{R}\) of the resultant force is \[ F_{R}=\int_{a}^{b}W(x)\, dx \]

The position \(\bar{x}\) of the resultant force measured in meters from the origin is given by \[ \overline{x}=\dfrac{\int_{a}^{b}xW(x)\, dx}{\int_{a}^{b}W(x)\, dx} \]

If the distributed force is \(W(x) =0.75x^{3},\) \(0\leq x\leq 5,\) find:

1. The magnitude of the resultant force.
2. The position from the origin of the resultant force.

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Chemistry: Enthalpy  In chemistry, enthalpy is a measure of the total energy of a system. For a nonreactive process with no phase change, the change in enthalpy \(\Delta H\) is given by \(\Delta H=\) \(\int_{T_{1}}^{T_{2}}C_{p}\, dT,\) where \(C_{p}\) is the specific heat of the system in question. The specific heat per mol of the chemical benzene is \[ C_{p} =0.126+(2.34\times 10^{-6}) T, \]

where \(C_{p}\) is in \(kJ/\left(mol {}^{\circ}{\rm C} \right)\), and \(T\) is in degrees Celsius.

1. What are the units of the change in enthalpy \(\Delta H?\)
2. What is the change in enthalpy \(\Delta H\) associated with increasing the temperature of 1.0 mol of benzene from \(20{}^{\circ}{\rm C}\) to \(40{}^{\circ}{\rm C}\)?
3. What is the change in enthalpy \(\Delta H\) associated with increasing the temperature of 1.0 mol of benzene from \(20{}^{\circ}{\rm C}\) to \(60{}^{\circ}{\rm C}\)?
4. Does the enthalpy of benzene increase, decrease, or remain constant as the temperature increases?

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Average Mass Density The mass density of a metal bar of length \(3\) meters is given by \(\rho (x) = 1000 + x - \sqrt{x}\) kilograms per cubic meter, where \(x\) is the distance in meters from one end of the bar. What is the average mass density over the length of the entire bar?

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Average Velocity The acceleration at time \(t\) of an object in rectilinear motion is given by \(a(t) =4\pi \cos t\). If the object's velocity is \(0\) at \(t = 0\), what is the average velocity of the object over the interval \(0\,\leq \,t\,\leq \,\pi \)?

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Average Area  What is the average area of all circles whose radii are between 1 and 3 m?

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Area

1. Use properties of integrals and the Fundamental Theorem of Calculus to find the area under the graph of \(y=3 - |x|\) from \(-3\) to \(3\).
2. Check your answer by using elementary geometry.

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Area

1. Use properties of integrals and the Fundamental Theorem of Calculus to find the area under the graph of \(y=1-\left\vert \dfrac{1}{2}x\right\vert \) from \(-2\) to \(2.\)
2. Check your answer by using elementary geometry. See the figure.

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Area  Let \(A\) be the area in the first quadrant that is enclosed by the graphs of \(y=3x^{2},\) \(y=\dfrac{3}{x},\) the \(x\)-axis, and the line \(x=k\), where \(k>1\), as shown in the figure.

1. Find the area \(A\) as a function of \(k\).
2. When the area is \(7\), what is \(k\)?
3. If the area \(A\) is increasing at the constant rate of \(5\) square units per second, at what rate is \(k\) increasing when \(k=15\)?

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Rectilinear Motion  A car starting from rest accelerates at the rate of \(3\) m./s\(^{2}\). Find its average speed over the first 8 seconds.

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Rectilinear Motion  A car moving at a constant velocity of 80 miles per hour begins to decelerate at the rate of 10 mi/h\(^{2}\). Find its average speed over the next 10 minutes.

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Average Slope

1. Use the definition of average value of a function to find the average slope of the graph of \(y=f(x)\), where \(a\leq x\leq b\). (Assume that \(f^\prime \) is continuous.)
2. Give a geometric interpretation.

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What theorem guarantees that the average slope found in Problem 91 is equal to \(f^{\prime }(u)\) for some \(u\) in \([a,b]?\) What different theorem guarantees the same thing? (Do you see the connection between these theorems?)

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Prove that if a function \(f\) is continuous on a closed interval \([a,b]\) and if \(k\) is a constant, then \(\int_{a}^{b}kf(x)\,dx=k \int_{a}^{b}f(x)\,dx\).

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Prove that if the functions \(f_{{1}}\), \(f_{{2}} , \ldots ,f_{n}\) are continuous on a closed interval \([a,b]\)and if \(k_{{1}}\), \(k_{{2}}, \ldots , k_{n}\) are constants, then \[ \begin{eqnarray*} &&\int_{a}^{b}[k_{{1}}f_{{1}}(x)+k_{{2}}f_{{2}}(x)+\cdots +k_{n}f_{n}(x)]\,dx \\ &&\enspace =k_{{1}}\int_{a}^{b}f_{{1}}(x)\,dx+k_{{2}}\int_{a}^{b}f_{{2}}(x)\,dx+\cdots + k_{n}\int_{a}^{b}f_{n}(x)\,dx \end{eqnarray*} \]

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Area The area under the graph of \(y=\cos x\) from \(-\dfrac{\pi }{2}\) to \(\dfrac{\pi }{2}\) is separated into two parts by the line \(x=k\), \(\dfrac{-\pi}{2}\lt k \lt \dfrac{\pi}{2}\), as shown in the figure. If the area under the graph of \(y\) from \(-\dfrac{\pi }{2}\) to \(k\) is three times the area under the graph of \(y\) from \(k\) to \(\dfrac{\pi }{2},\) find \(k.\)

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Displacement of a Damped Spring The displacement \(x\) in meters of a damped spring from its equilibrium position at time \(t\) seconds is given by \[ x(t) =\dfrac{\sqrt{15}}{10}e^{-t}\sin \big(\sqrt{15}t\big) + \dfrac{3}{2}e^{-t}\cos \big(\sqrt{15}t\big) \]

1. What is the displacement of the spring at \(t=0?\)
2. Graph the displacement for the first \(2\) seconds of the springs' motion.
3. Find the average displacement of the spring for the first 2 seconds of its motion.

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Area Let \[ f(x) =\vert x^{4}+3.44x^{3}-0.5041x^{2}-5.0882x +\,1.1523 \vert \]

be defined on the interval \([-3,1] \). Find the area under the graph of \(f.\)

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If \(f\) is continuous on \([a,b]\), show that the functions defined by \[ F(x)= \int_{c}^{x}{f(t)\,dt} \qquad G(x)= \int_{d}^{x}{f(t)\,dt} \]

for any choice of \(c\) and \(d\) in \((a,b)\) always differ by a constant. Also show that \[ F(x)-G(x)={\int_{c}^{d}{f(t)\,dt}} \]

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Put It Together Suppose \(a\lt c\lt b\) and the function \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\). Which of the following is not necessarily true?

1. \(\int_{a}^{b}f(x)\,dx= \int_{a}^{c}{f(x)\,dx}+\int_{c}^{b}f(x)\,dx\).
2. There is a number \(d\) in \((a,b)\) for which \[ f^\prime (d)= \dfrac{f(b)-f(a)}{b-a} \]
3. \(\int_{a}^{b}{f(x)\,dx\geq 0}\).
4. \(\lim\limits_{x\rightarrow c}\,f(x)=f(c)\).
5. If \(k\) is a real number, then \(\int_{a}^{b}kf(x)\,dx=k \int_{a}^{b}f(x)\,dx.\)

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Minimizing Area  Find \(b>0\) so that the area enclosed in the first quadrant by the graph of \(y=1+b-bx^2\) and the coordinate axes is a minimum.

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Area  Find the area enclosed by the graph of \(\sqrt{x}+\sqrt{y}=1\) and the coordinate axes.

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Average Speed  For a freely falling object starting from rest, \(v_{0}=0,\) find:

1. The average speed \(\overline{v }_{t}\) with respect to the time \(t\) in seconds over the closed interval \([0,5].\)
2. The average speed \(\overline{v }_{s}\) with respect to the distance \(s\) of the object from its position at \(t=0\) over the closed interval \([0,s_{1}] \), where \(s_{1}\) is the distance the object falls from \(t=0\) to \(t=5\) seconds.

(Hint: The derivation of the formulas for freely falling objects is given in Section 4.8, pp. 334-337.)

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Average Speed If an object falls from rest for 3 s, find:

1. Its average speed with respect to time.
2. Its average speed with respect to the distance it travels in 3 seconds.

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Free Fall  For a freely falling object starting from rest, \(v_{0}=0\), find:

1. The average velocity \(\overline{v }_{t}\) with respect to the time \(t\) over the closed interval \([0,t_{1}].\)
2. The average velocity \(\overline{v }_{s}\) with respect to the distance \(s\) of the object from its position at \(t=0\) over the closed interval \([0,s_{1}]\), where \(s_{1}\) is the distance the object falls in time \(t_{1}\). Assume \(s(0) =0.\)

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Put It Together

1. What is the domain of \(f(x) =2 |x - 1| x^{2}\)?
2. What is the range of \(f\)?
3. For what values of \(x\) is \(f\) continuous?
4. For what values of \(x\) is the derivative of \(f\) continuous?
5. Find \(\int_{0}^{1}{f(x)\,dx.}\)

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Probability  A function \(f\) that is continuous on the closed interval \([a,b],\) and for which (i) \(f(x)\geq 0\) for numbers \(x\) in \([a,b]\) and 0 elsewhere and (ii) \(\int_{a}^{b} f (x)\,dx=1,\) is called a probability density function. If \(a\leq c\lt d\leq b\), the probability of obtaining a value between \(c\) and \(d\) is defined as \(\int_{c}^{d}f(x)\,dx\).

1. Find a constant \(k\) so that \(f(x)=kx\) is a probability density function on \([0,2]\).
2. Find the probability of obtaining a value between \(1\) and \(1.5\).

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Cumulative Probability Distribution  Refer to Problem 106. If \(f\) is a probability density function the cumulative distribution function \(F\) for \(f\) is defined as \[ F(x)=\int_{a}^{x}f(t)\,dt \qquad a\leq x\leq b \]

Find the cumulative distribution function \(F\) for the probability density function \(f(x) =kx\) of Problem 106(a).

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For the cumulative distribution function \(F(x)=x-1\), on the interval \([1,2]\):

1. Find the probability density function \(f\) corresponding to \(F\).
2. Find the probability of obtaining a value between \(1.5\) and \(1.7\).

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Let \(f(x)=x^{3}-6x^{2}+11x-6\). Find \(\int_{1}^{3}\vert f(x)\vert dx.\)

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Show that for \(x>1,\) \(\ln x\lt2 (\sqrt{x}-1).\)

(Hint: Use the result given on Problem 114.)

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Prove that the average value of a line segment \(y=m (x-x_{1} ) +y_{1}\) on the interval \([x_{1},x_{2}]\) equals the \(y\)-coordinate of the midpoint of the line segment from \(x_{1}\) to \(x_{2}.\)

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Prove that if a function \(f\) is continuous on a closed interval \([a,b]\) and if \(f(x)\geq 0\) on \([a,b]\), then \(\int_{a}^{b}f(x)\,dx \geq 0\).

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1. Prove that if \(f\) is continuous on a closed interval \([a,b]\) and \({\int_{a}^{b}{f(x)\,dx=0}}\), there is at least one number \(c\) in \([a,b]\) for which \(f(c) = 0\).
2. Give a counterexample to the statement above if \(f\) is not required to be continuous.

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Prove that if functions \(f\) and \(g\) are continuous on a closed interval \([a,b]\) and if \(f(x)\geq \,g(x)\) on \([a,b]\), then \[ \int_{a}^{b}f(x)\,dx\geq \int_{a}^{b}g(x)\,dx. \]

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Prove that if \(f\) is continuous on \([a,b]\), then \[ \left\vert \int_{a}^{b} f(x)\,dx\right\vert \leq \int_{a}^{b}\vert {f(x)}\vert\, dx. \]

Give a geometric interpretation of the inequality.