The Integral

Chapter Review

THINGS TO KNOW

5.1 Area

Definitions:

Partition of an interval \([a,b]\) (p. 345)
Area \(A\) under the graph of a function \(f\) from \(a\) to \(b\) (p. 348)

5.2 The Definite Integral

Definitions:

Riemann sums (pp. 353–354)
The definite integral (p. 355)
\(\int_{a}^{a}f(x)\,dx=0\) (p. 356)
\(\int_{a}^{b}f(x)\,dx=-\int_{b}^{a}f(x)\,dx\) (p. 356)

Theorems:

If a function \(f\) is continuous on a closed interval \([a,b]\),then the definite integral\( \int_{a}^{b}f(x)\,dx\) exists. (p. 356)
\(\int^b_a h\ dx = h(b-a)\), \(h\) a constant (p. 357)

5.3 The Fundamental Theorem of Calculus

Fundamental Theorem of Calculus: Let \(f\) be a function that is continuous on a closed interval \([a,b]\).

Part 1: The function \(I\) defined by \(I\,(x)=\int_{a}^{x}f(t)\,dt\) has the properties that it is continuous on \([a,b] \) and differentiable on \((a,b).\) Moreover, \(I^\prime (x)=\dfrac{d}{dx} \left[ \int_{a}^{x}f(t)\,dt\right] =f(x),\) for all \(x\) in \((a,b).\) (p. 362)
Part 2: If \(F\) is any antiderivative of \(f\) on [\(a,b\)], then \(\int_{a}^{b}f(x)\,dx=F(b)-F(a)\). (p. 364)

5.4 Properties of the Definite Integral

Properties of definite integrals:

If two functions \(f\) and \(g\) are continuous on the closed interval \([a,b]\) and \(k\) is a constant, then

Integral of a sum: \[\int_{a}^{b}\,[ f(x)+g(x)] \,dx=\int_{a}^{b}f(x)\,dx + \int_{a}^{b}g(x)\,dx\,\,\text{(p. 369)}\]
Integral of a constant times a function:\[\int_{a}^{b}kf(x)\,dx = k\int_{a}^{b}f(x)\,dx\,\,\text{(p. 369)}\]
\(\int_{a}^{b}\left[ k_{{\rm 1}}f_{{\rm 1}}(x)+k_{{\rm 2}}f_{{\rm 2}}(x)+\cdots +\,k_{n} f_{n}(x)\right] \, dx = k_{1}\int_{a}^{b} f_{1}(x)\,dx + k_{2}\int_{a}^{b}f_{2}(x)\,dx+\cdots + k_{n}\int_{a}^{b}f_{n}(x)\,dx\,\,\text{. (p. 370)}\)
If \(f\) is continuous on an interval containing the numbers \(a\), \(b\), and \(c\), then \[\int_{a}^{b}f(x)\,dx=\int_{a}^{c}f(x)\,dx+\int_{c}^{b}f(x)\,dx\,\,\text{(p. 370)}\]
Bounds on an Integral: If a function \(f\) is continuous ona closed interval \([a,b]\) and if \(m\) and \(M\) denote the absolute minimum andabsolute maximum values, respectively, of \(f\) on \([a,b]\), then\[m(b-a)\leq \int_{a}^{b}f(x)\,dx\leq M\,(b-a)\,\,\text{(p. 371)}\]
Mean Value Theorem for Integrals: If a function \(f\) is continuous on a closed interval \([a,b]\), then there is a real number \(u\), where \(a\,\leq \,u\,\leq \,b\), for which\[\int_{a}^{b}f(x)\,dx = f(u)(b-a)\,\,\text{(p. 372)}\]

Definition: The average value of a function over an interval \([a,b] \) is \(\bar{y}=\dfrac{1}{b-a}\int_{a}^{b}f(x)\,dx\) (p. 374)

5.5 The Indefinite Integral; Growth and Decay Models

The indefinite integral of \(f\): \(\int f(x)\,dx=F(x) +C\) if and only if \(\dfrac{d}{dx}[ F(x) +C] =f(x),\) where \(C\) is the constant ofintegration. (p. 379)

Basic integration formulas: See Table #. (p. 380)

Properties of indefinite integrals:

Derivative of an integral:\[\dfrac{d}{dx}\left[ \int f(x)\,dx\right]=f(x)\,\,\text{(p. 380)}\]
Integral of a sum:\[\int\, [ f(x)+g(x)] dx=\intf(x)\,dx +\int g(x)dx\,\,\text{(p. 381)}\]
Integral of a constant \(k\) times a function:\[\int k\,f(x)\,dx =k\int f(x)\,dx\,\,\text{(p. 381)}\]

5.6 Method of Substitution; Newton's Law of Cooling

Method of substitution: (p. 388)

Method of substitution (definite integrals):

Find the related indefinite integral using substitution. Then use theFundamental Theorem of Calculus. (p. 391)
Find the definite integral directly by making a substitution in the integrand and using the substitution to change the limits of integration. (p. 391)

Basic integration formulas:

\(\int \dfrac{g^\prime (x) }{g(x) }dx\) \(=\ln \vert g(x)\vert +C\) (p. 390)
If \(f\) is an even function, then \[\int_{-a}^{a}f(x)\,dx=2\int_{{0}}^{a}f(x)\,dx\,\,\text{(p. 393)}\]
If \(f\) is an odd function, then \({\int_{-a}^{a}}f(x)\,dx=0\). (p. 393)

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OBJECTIVES

Section	You should be able to …	Examples	Review Exercises
5.1	1 Approximate the area under the graph of a function (p. 344)	1, 2	1, 2
	2 Find the area under the graph of a function (p. 348)	3, 4	3, 4
5.2	1 Define a definite integral as the limit of Riemann sums (p. 353)	1, 2	5(a), (b)
	2 Find a definite integral using the limit of Riemann sums (p. 356)	3–5	5(c)
5.3	1 Use Part 1 of the Fundamental Theorem of Calculus (p. 363)	1–3	7-10, 52, 53
	2 Use Part 2 of the Fundamental Theorem of Calculus (p. 365)	4, 5	5(d), 11–13, 15–18, 56
	3 Interpret an integral using Part 2 of the Fundamental Theorem of Calculus (p. 365)	6	6–21, 22, 57
5.4	1 Use properties of the definite integral (p. 369)	1–6	23, 24, 27, 28, 53
	2 Work with the Mean Value Theorem for Integrals (p. 372)	7	29, 30
	3 Find the average value of a function (p. 373)	8	31–34
5.5	1 Find indefinite integrals (p. 379)	1	14
	2 Use properties of indefinite integrals (p. 380)	2, 3	19, 20, 35, 36
	3 Solve differential equations involving growth and decay (p. 382)	4, 5	37, 38, 59, 60
5.6	1 Find an indefinite integral using substitution (p. 387)	1–5	39–41, 44, 45, 48, 51
	2 Find a definite integral using substitution (p. 391)	6, 7	42, 43, 46, 47, 50, 51, 54, 55, 49
	3 Integrate even and odd functions (p. 393)	8, 9	25–26
	4 Solve differential equations: Newton's Law of Cooling (p. 394)	10	58

REVIEW EXERCISES

Question 5.518

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 \(n = 8\).

Question 5.519

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.

Question 5.520

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 \(f)\).

Question 5.521

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 \(f).\)

Question 5.522

Riemann Sums

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

Question 5.523

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

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

Question 5.524

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

Question 5.525

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

Question 5.526

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

Question 5.527

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

In Problems 11–20, find each integral.

Question 5.528

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

Question 5.529

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

Question 5.530

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

Question 5.531

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

Question 5.532

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

Question 5.533

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

Question 5.534

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

Question 5.535

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

Question 5.536

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

Question 5.537

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

Question 5.538

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

Question 5.539

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 \(\int_{0}^{2}f(t) dt=100.\)

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In Problems 23–26, find each integral.

Question 5.540

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

Question 5.541

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

Question 5.542

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

Question 5.543

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

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

Question 5.544

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

Question 5.545

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

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

Question 5.546

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

Question 5.547

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

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

Question 5.548

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

Question 5.549

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

Question 5.550

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

Question 5.551

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

Question 5.552

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

Question 5.553

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

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

Question 5.554

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

Question 5.555

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

In Problems 39–51, find each integral.

Question 5.556

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

Question 5.557

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

Question 5.558

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

Question 5.559

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

Question 5.560

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

Question 5.561

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

Question 5.562

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

Question 5.563

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

Question 5.564

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

Question 5.565

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

Question 5.566

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

Question 5.567

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

Question 5.568

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

Question 5.569

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

Question 5.570

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.

Question 5.571

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

Question 5.572

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

Question 5.573

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

Question 5.574

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 gatehas been opened. If the gate is opened at 7 a.m. and is left open until 9:24a.m., how much water is released?

Question 5.575

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 thebody with respect time \(t\) inhour (h) is proportional to \(u(t)-T\), where \(T\) is the ambient temperature and the constant ofproportionality is \(-0.294\).

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

Question 5.576

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

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

Question 5.577

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

Write a differential equation that models the growth rate of the population.
Find the general solution of the differential equation.
Find the particular solution of the differential equationif in 2010 (\(t=0\)), the population of China was \(1.341335\times10^{9}.\)
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.

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