REVIEW EXERCISES

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Area  Approximate the area under the graph of \(f(x) =2x+1\) from \(0\) to \(4\) by finding \(s_{n}\) and \(S_{n}\) for \(n = 4\) and \(n = 8\).

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Area  Approximate the area under the graph of \(f(x) =x^{2}\) from 0 to 8 by finding \(s_{n}\) and \(S_{n}\) for \(n = 4\) and \(n = 8\) subintervals.

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Area  Find the area \(A\) under the graph of \(y = f(x) = 9 - x^{2}\) from \(0\) to \(3\) by using lower sums \(s_{n}\) (rectangles that lie below the graph of \(f)\).

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Area  Find the area \(A\) under the graph of \(y = f(x) = 8-2x\) from \(0\) to \(4\) using upper sums \(S_{n}\) (rectangles that lie above the graph of \(f).\)

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Riemann Sums

  1. Find the Riemann sum of \(f(x) =x^{2}-3x+3\) on the closed interval \([-1,3] \) using a regular partition with four subintervals and the numbers \(u_{1}=-1,\) \(u_{2}=0,\) \(u_{3}=2,\) and \(u_{4}=3.\)
  2. Find the Riemann sums of \(f\) by partitioning \([-1,3] \) into \(n\) subintervals of equal length and choosing \(u_{i}\) as the right endpoint of the \(i\)th subinterval \([x_{i-1},x_{i}]\). Write the limit of the Riemann sums as a definite integral. Do not evaluate.
  3. Find the limit as \(n\) approaches \(\infty \) of the Riemann sums found in (b).
  4. Find the definite integral from (b) using the Fundamental Theorem of Calculus. Compare the answer to the limit found in (c).

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Units of an Integral  In the definite integral \(\int_{a}^{b}a(t)\,dt,\) where \(a\) represents acceleration measured in meters per second squared and \(t\) is measured in seconds, what are the units of \(\int_{a}^{b}a(t)\, dt?\)

In Problems 7–10, find each derivative using the Fundamental Theorem of Calculus.

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\(\dfrac{d}{dx}\int_{0}^{x}t^{2/3}\sin t\) \(dt\)

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\(\dfrac{d}{dx}\int_{e}^{x}\ln t\, dt\)

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\(\dfrac{d}{dx}\int_{x^{2}}^{1}\tan\, t\,dt\)

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\(\dfrac{d}{dx}\int_{a}^{2\sqrt{x}}\dfrac{t}{t^{2}+1}\,dt\)

In Problems 11–20, find each integral.

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

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

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

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

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

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

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

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

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\(\int \left( \dfrac{1+2xe^{x}}{x}\right)\, dx\)

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\(\int \dfrac{1}{2}\sin x~dx \)

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Interpreting an Integral  The function \(v=v(t) \) is the speed \(v,\) in kilometers per hour, of a train at a time \(t\), in hours. Interpret the integral \(\int_{0}^{16}v(t)\,dt=460.\)

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Interpreting an Integral  The function \(V=f(t) \) is the volume \(V\) of oil, in liters per hour, draining from a storage tank at time \(t\) (in hours). Interpret the integral \(\int_{0}^{2}f(t) dt=100.\)

402

In Problems 23–26, find each integral.

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

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

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

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

Bounds on an IntegralIn Problems 27 and 28, find lower and upper bounds for each integral.

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

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

In Problems 29 and 30, for each integral find the number(s) \(u\) guaranteed by the Mean Value Theorem for Integrals.

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

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

In Problems 31–34, find the average value of each function over the given interval.

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

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\(f(x) =x^{3}\) over \([1,4] \)

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

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\(f(x) =6x^{2/3}\) over \([ 0, 8 ] \)

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Find \(\dfrac{d}{dx}\int \sqrt{\dfrac{1}{1+4x^{2}}}\,dx\)

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Find \(\dfrac{d}{dx}\int\, \ln\, x\, dx\).

In Problems 37 and 38, solve each differential equation using the given boundary condition.

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\(\dfrac{dy}{dx}=3xy\);  \(y=4\) when \(x=0\)

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\(\cos y\dfrac{dy}{dx}=\dfrac{\sin y}{x}\); \(y=\dfrac{\pi }{3}\) when \(x=-1 \)

In Problems 39–51, find each integral.

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\(\int {{\dfrac{{y\,dy}}{{(y-2)^{3}}}}}\)

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\(\int {{\dfrac{{x}}{{(2-3x)}^{3}}}\,dx}\)

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\(\int {\sqrt{\dfrac{{1+x}}{{x^{5}}}}\,dx} \), \(x>0\)

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\(\int_{\pi ^{2}/4}^{4\pi ^{2}}\dfrac{1}{\sqrt{x}}\sin \sqrt{x}~dx\)

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\(\int_{1}^{2}\dfrac{1}{t^{4}}\left( 1-\dfrac{1}{t^{3}}\right) ^{3}~dt\)

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\(\int \dfrac{e^{x}+1}{e^{x}-1}dx\)

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\(\int \dfrac{dx}{\sqrt{x}\,( 1-2\sqrt{x}) }\)

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

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\(\int_{-1}^{1}\dfrac{5^{-x}}{2^{x}}dx\)

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\(\int e^{x+e^{x}}dx\)

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\(\int_{0}^{1}\dfrac{x\,dx}{\sqrt{2-x^{4}}}\)

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\(\int_{4}^{5}\dfrac{dx}{x\sqrt{x^{2}-9}}\)

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\(\int {\sqrt[3]{{x^{3}+3\cos x}} (x^{2}-\sin x)\,dx}\)

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Find \(f^{\prime \prime} (x)\) if \(f(x)={\int_{0}^{x}\sqrt{1-t^{2}}}d{t}\).

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Suppose that \(F(x)={\int_{0}^{x}{\sqrt{t}\,dt}}\) and \(G(x)={\int_{1}^{x}\sqrt{t}}\,dt\). Explain why \(F(x)-G(x)\) is constant. Find the constant.

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If \(\int_{0}^{2}f(x + 2)\,dx\,=3\), find \({\int_{2}^{4}{f(x)\,dx.}}\)

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If \(\int_{1}^{2}f(x - c)\,dx=5\), where \(c\) is a constant, find \(\int_{1-c}^{2-c}f(x)\,dx\).

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Area  Find the area under the graph of \(y=\cosh x\) from \(x=0\) to \(x=2\).

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Water Supply  A sluice gate of a dam is opened and water is released from the reservoir at a rate of \(r(t) =100+\sqrt{t}\) gallons per minute, where \(t\) measures the time in minutes since the gate has been opened. If the gate is opened at 7 a.m. and is left open until 9:24 a.m., how much water is released?

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Forensic Science  A body was found in a meat locker whose ambient temperature is \(10{}^{\circ}{\rm C}.\) When the person was alive, his body temperature was \(37{}^{\circ}{\rm C}\) and now it is \(25{}^{\circ}{\rm C}.\) Suppose the rate of change of the temperature \(u=u(t)\) of the body with respect time \(t\) in hour (h) is proportional to \(u(t)-T\), where \(T\) is the ambient temperature and the constant of proportionality is \(-0.294\).

  1. Write a differential equation that models the temperature \(u=u(t) \) of the body at time \(t\).
  2. Find the general solution of the differential equation.
  3. Find the particular solution of the differential equation, using the initial condition that at the time of death, \(u(0) =37{}^{\circ}{\rm C}.\)
  4. If the body was found at \(1\)a.m., when was the murder committed?
  5. How long will it take for the body to cool to \(12{}^{\circ}{\rm C}?\)

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Radioactive Decay  The amount \(A\) of the radioactive element radium in a sample decays at a rate proportional to the amount of radium present. Given the half-life of radium is \(1690\) years:

  1. Write a differential equation that models the amount \(A\) of radium present at time \(t\).
  2. Find the general solution of the differential equation.
  3. Find the particular solution of the differential equation with the initial condition \(A(0) =10 g.\)
  4. How much radium will be present in the sample at \(t=300\) years?

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National Population Growth  Barring disasters (human-made or natural), the population \(P\) of humans grows at a rate proportional to its current size. According to the U.N. World Population studies, from 2005 to 2010 the population of China grew at an annual rate of \(0.510\%\) per year.

  1. Write a differential equation that models the growth rate of the population.
  2. Find the general solution of the differential equation.
  3. Find the particular solution of the differential equation if in 2010 (\(t=0\)), the population of China was \(1.341335\times 10^{9}.\)
  4. If the rate of growth continues to follow this model, when will the projected population of China reach \(2\) billion persons?

Source: U.N. World Population Prospects, 2010 update.