5.7 Properties of the Definite Integral

OBJECTIVES

When you finish this section, you should be able to:

  1. Use properties of the definite integral (p. 369)
  2. Work with the Mean Value Theorem for Integrals (p. 372)
  3. Find the average value of a function (p. 373)

We have seen that there are properties of limits that make it easier to findlimits, properties of continuity that make it easier to determinecontinuity, and properties of derivatives that make it easier to findderivatives. Here, we investigate several properties of the definite integralthat will make it easier to find integrals.

5.7.1 Use Properties of the Definite Integral

Proofs of most of the properties in this section require the use of thedefinition of the definite integral. However, if the condition is added thatthe integrand is continuous, then the Fundamental Theorem ofCalculus can be used to establish the properties. We will use this addedcondition in the proofs included here.

NOTE

From now on, we will refer to Parts 1 and 2 of the FundamentalTheorem of Calculus simply as the Fundamental Theorem ofCalculus.

THEOREM The Integral of the Sum of Two Functions

If two functions \(f\) and \(g\) are continuous on the closedinterval \([a,b]\), then\[\begin{equation*}\boxed{\bbox[5pt]{\int_{a}^{b}[ f(x)+g(x)] \,dx=\int_{a}^{b}f(x)\,dx+\int_{a}^{b}g(x)\,dx }}\tag{1}\end{equation*}\]

IN WORDS

The integral of a sum equals the sum of the integrals.

Proof

Since \(f\) and \(g\) are continuous on \([a,b],\) then the definiteintegral of each function exists. Let \(F\) and\(G\) be an antiderivative of \(f\) and \(g,\) respectively, on\((a,b) \). Then \(F' =f\) and \(G^{\prime} =g\) on \((a,b) \).

Also, since \((F + G)' = F^\prime + G' = f + g,\)then \(F+G\) is an antiderivative of \(f+g\) on \((a,b) .\) Now\[\begin{eqnarray*}\int_{a}^{b}[f(x)+g(x)]\,dx=\bigg[F(x)+G(x)\bigg] _{a}^{b} &=&[F(b)+G(b)]-[F(a)+G(a)] \\&=&[F(b)-F(a)]+[G(b)-G(a)] \\&=&{\int_{a}^{b}{f(x)\,dx+{\int_{a}^{b}{g(x)\,dx}}}} \end{eqnarray*}\]

EXAMPLE 1 Using Property (1) of the Definite Integral

\[\begin{eqnarray*}\int_{0}^{1}(x^{2}+e^{x})\,dx &=& \int_{0}^{1}x^{2}dx+\int_{0}^{1}e^{x}\,dx=\left[ \dfrac{x^{3}}{3}\right] _{0}^{1}+ \Big[ e^{x}\Big] _{0}^{1} \\[4pt]&=&\left[ \dfrac{1^{3}}{3}-0\right] +\Big[ e^{1}-e^{0}\Big] =\dfrac{1}{3}+e-1=e-\dfrac{2}{3} \end{eqnarray*}\]

NOW WORK

Problem 13.

THEOREM The Integral of a Constant Times a Function

Suppose a function \(f\) is continuous on the closed interval \([a,b]\). If \(k\) is a constant, then\[\begin{equation*}\boxed{\bbox[5pt]{\int_{a}^{b}kf(x)\,dx=k\int_{a}^{b}f(x)\,dx }}\tag{2}\end{equation*}\]

IN WORDS

A constant factor can be factored out of an integral.

You are asked to prove this theorem in Problem 93.

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EXAMPLE 2 Using Property (2) of the Definite Integral

\[\begin{eqnarray*}{\int_{1}^{e}}\dfrac{{3}}{x}{\,dx=3} {{\,{\int_{1}^{e}}}}\dfrac{1}{x}{dx=3} \Big[ \ln \vert x\vert \Big] _{1}^{e}={3}\left( {\ln \,}e-\ln1\right) =3(1-0) =3 \end{eqnarray*}\]

NOW WORK

Problem 15.

The two properties above can be extended as follows:

THEOREM

Suppose each of the functions \(f_{{1}}\), \(f_{{2}}, \ldots , f_{n}\) is continuous on the closed interval \([a,b]\). If \(k_{{1}}\), \(k_{{2}}, \ldots\), \(k_{n}\) are constants, then \[\begin{equation*}\boxed{\bbox[5pt]{\begin{array}{@{\hspace*{-6pt}}lll@{}}&&\int_{a}^{b}\left[ k_{{1}}f_{{1}}(x)+k_{{2}}f_{{2}}(x)+\cdots+\,k_{n} f_{n}(x)\right]\,dx \\[9pt]&&  = k_{{1}}\int_{a}^{b}f_{{1}}(x)\,dx + k_{{2}} \int_{a}^{b}f_{{2}}(x)\,dx+\cdots + k_{n}\int_{a}^{b}f_{n}(x)\,dx \end{array}}}\tag{3}\end{equation*}\]

You are asked to prove this theorem in Problem 94.

EXAMPLE 3 Using Property (3) of the Definite Integral

Find \(\int_{1}^{2}\dfrac{3x^{3}-6x^{2}-5x+4}{2x}dx\).

Solution The function \(f(x)=\dfrac{3x^{3}-6x^{2}-5x+4}{2x}\) is continuous on the closed interval \([1,2] \). Using algebra and the properties of the definite integral, we get\[\begin{eqnarray*}\int_{1}^{2}\dfrac{3x^{3}-6x^{2}-5x+4}{2x}dx &=&\int_{1}^{2}\left[\dfrac{3}{2} x^{2}- 3x- \dfrac{5}{2}+\dfrac{2}{x}\right] dx\\&=& \int_{1}^{2} \dfrac{3}{2} x^{2}dx-\int_{1}^{2} 3x\, dx-\int_{1}^{2} \dfrac{5}{2} \, dx+\int_{1}^{2}\dfrac{2}{x}~dx \\[4pt]&=& \dfrac{3}{2}\int_{1}^{2}x^{2}~dx-3\int_{1}^{2}x\, dx-\dfrac{5}{2}\int_{1}^{2}~dx+2\int_{1}^{2}\dfrac{1}{x}~dx \\&=& \dfrac{3}{2} \left[ \dfrac{x^{3}}{3}\right] _{1}^{2} -3 \left[ \dfrac{x^{2}}{2}\right] _{1}^{2} -\dfrac{5}{2} \Big[x\Big]_{1}^{2} +2 \Big[\ln \vert x\vert \Big]_{1}^{2} \\&=&\dfrac{1}{2} ( 8-1) - 3\left( 2-\dfrac{1}{2}\right)-\dfrac{5}{2}(2-1) + 2( \ln 2- \ln 1 ) \\&=&-\dfrac{7}{2}+2\ln2\approx -2.114\end{eqnarray*}\]

NOW WORK

Problem 27.

The next property states that a definite integral of a function \(f\)from \(a \) to \(b\) can be evaluated in pieces.

THEOREM

If a function \(f\) is continuous on an interval containing thenumbers \(a\), \(b\), and \(c\), then \[\begin{equation*}\boxed{\bbox[5pt]{\int_{a}^{b}f(x)\,dx=\int_{a}^{c}f(x)\,dx+\int_{c}^{b}f(x)\,dx }}\tag{4}\end{equation*}\]

A proof of this theorem is given in Appendix B.

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In particular, if \(f\) is continuous and nonnegative on a closedinterval \([a,b]\) and if \(c\) is a number between \(a\) and \(b\), then this property has asimple geometric interpretation, as seen in Figure 5.24.

EXAMPLE 4 Using Property (4) of the Definite Integral

  1. If \(f\) is continuous on the closed interval \([2,7]\), then \[\int_{2}^{7}f(x)\,dx=\int_{2}^{4}f(x)\,dx+\int_{4}^{7}f(x)\,dx \]
  2. If \(g\) is continuous on the closed interval \([3,25]\), then \[\int_{3}^{10}g(x)\,dx=\int_{3}^{25}g (x)\,dx+ \int_{25}^{10}g(x) dx \]

Example 4(b) illustrates that the number \(c\) need not lie between\(a\) and \(b\).

NOW WORK

Problem 43.

Property (4) is useful when integrating piecewise-defined functions.

EXAMPLE 5 Using Property (4) of the Definite Integral

Find the area \(A\) under the graph of\[f(x) =\left\{ \begin{array}{c@{ }c@{ }ccc}x^{2} & \hbox{if} & 0 & \leq x &\lt10 \\ 100 & \hbox{if} & 10 &\leq x & \leq 15\end{array}\right.\]

from \(0\) to \(15.\)

Solution See Figure 5.25. Since \(f\) is nonnegative on the closed interval\([0,15] ,\) then \(\int_{0}^{15}f(x)\, dx\) equals the area \(A\) under the graph of \(f\)from \(0\) to \(15.\) Since \(f\) is continuous on \([0,15],\)\[\begin{eqnarray*}\int_{0}^{15}f(x)\, dx &=& \int_{0}^{10}f(x)\, dx+\int_{10}^{15}f(x)\,dx=\int_{0}^{10}x^{2}dx+\int_{10}^{15}100\,dx \\[4.5pt]&=& \left[\dfrac{x^{3}}{3}\right] _{0}^{10}+ \Big[100x\Big]_{10}^{15}=\dfrac{1000}{3}+500=\dfrac{2500}{3}\end{eqnarray*}\]

Figure 5.25: \(A=\int^{15}_0 f(x)\, dx\)

The area under the graph of \(f\) is approximately 833.33 square units.

NOW WORK

Problem 35.

The next property establishes bounds on a definite integral.

THEOREM Bounds on an Integral

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\[\begin{equation*}\boxed{\bbox[5pt]{m(b-a)\leq \int_{a}^{b}f(x)\,dx\leq M\,(b-a) }}\tag{5}\end{equation*}\]

A proof of this theorem is given in Appendix B.

If \(f\) is nonnegative on \([a,b]\), then the inequalities in (5) have ageometric interpretation. In Figure 5.26, the area of the shadedregion is \({\int_{a}^{b}{f(x)\,dx}}\). The smaller rectangle has width\(b-a,\) height\(\, m\), and area equal \(m(b-a).\) The larger rectangle has width \(b-a,\)height \(M\), and area \(M(b-a)\). These three areas are numerically related by theinequalities in the theorem.

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EXAMPLE 6 Using Property (5) of Definite Integrals

  1. Find an upper estimate and a lower estimate for the area \(A\) under thegraph of \(f(x) =\sin x\) from \(0\) to \(\pi \).
  2. Find the actual area under the graph.

Solution The graph of \(f\) is shown in Figure 5.27.Since \(f(x) \geq 0\) for all \(x\) in the closed interval \([ 0,\pi ] \), the area \(A\) under its graph is given by thedefinite integral, \(\int_{0}^{\pi }\sin x~dx\).

Figure 5.27: \(f(x) = \sin x, 0 \leq x \leq \pi\).

NEED TO REVIEW?

The Extreme Value Theorem is discussed in Section 4.2, pp. xx-xx.

(a) From the Extreme Value Theorem, \(f\) has anabsolute minimum value and an absolute maximum value on the interval \([0,\pi] .\)The absolute maximum of \(f\) occurs at \(x=\dfrac{\pi}{2},\) and its value is \(f \left( \dfrac{\pi }{2}\right) =\sin \dfrac{\pi }{2}=1.\) The absoluteminimum occurs at \(x=0\) and at \(x=\pi \); the absolute minimumvalue is \(f (0) =\sin 0=0=f (\pi)\). Using the inequalities in (5), the area under the graph of \(f\) is bounded as follows:\[0\leq \int_{0}^{\pi }\sin x~dx\leq \pi \]

(b) The actual area under the graph is\[A= \int_{0}^{\pi}\sin x\,dx= \bigg[-\cos x\bigg] _{0}^{\pi }=-\cos \pi+\cos 0=1+1=2\hbox{ square units} \]

NOW WORK

Problem 53.

5.7.2 Work with the Mean Value Theorem for Integrals

Suppose \(f\) is a function that is continuous and nonnegative on a closedinterval \([a,b] .\) Figure 5.28 suggests that the area underthe graph of \(f\) from \(a\) to \(b\), \(\int_{a}^{b}f(x)\,dx\), is equal to the area of some rectangle ofwidth \(b-a\) and height \(f(u)\) for a some choice (orchoices) of \(u\) in the interval \([a,b].\)

Figure 5.28: \(\int^b_a f(x) \, dx =f(u) (b-a) \).

In more general terms, for every function \(f\) that is continuous on a closedinterval \([a,b] ,\) there is some number \(u\) (not necessarilyunique) in the interval \([a,b] \) for which \[\int_{a}^{b}f(x)\, dx=f(u) (b-a)\]

This result is known as the Mean Value Theorem forIntegrals.

THEOREM Mean Value Theorem for Integrals

If a function \(f\) is continuous on a closed interval \([a,b]\), thereis a real number \(u\), \(a\,\leq \,u\,\leq \,b\), for which \[\boxed{\bbox[5pt]{\int_{a}^{b}f(x)\,dx=f(u)(b-a) }}\tag{6}\]

Proof

Let \(f\) be a function that is continuous on a closed interval \([a,b] \).

If \(f\) is a constant function, say, \(f(x) =k,\) on \([a,b]\), then \[\int_{a}^{b}f(x)\,dx= \int_{a}^{b}k\,dx =k(b-a) =f(u)(b-a) \]

for any choice of \(u\) in \([a,b]\).

If \(f\) is not a constant function on \([a,b]\), then by the Extreme ValueTheorem, \(f\) has an absolute maximum and an absolute minimum on \([a,b]\). Suppose \(f\) assumes its absolute minimum at the number \(c\) so that \(f(c)=m\); and suppose \(f\) assumes its absolute maximum at the number \(C\)so that \(f (C)\,=M\). Then by the Bounds on an Integral Theorem (5), we have\[m(b-a)\leq {\int_{a}^{b}{f(x)\,dx\leq M(b-a)}}\qquad \hbox{for all } x\hbox{ in }[a,b]\]

373

We divide each part by \((b-a) \) and replace \(m\) by \(f(c)\) and\(M\) by \(f (C)\). Then \[f(c)\leq \dfrac{{1}}{{b-a}}\int_{a}^{b}f(x)\,dx\leq f(C)\]

NEED TO REVIEW?

The Intermediate Value Theorem is discussed in Section 1.3, pp. xx-xx.

Since \(\dfrac{1}{b-a}\int_{a}^{b}f(x)\,dx\) is a real number between \(f(c)\) and \(f (C)\), it follows from the Intermediate Value Theorem that there is a real number \(u\) between \(c\) and \(C\), for which \[f(u)\,={\dfrac{{1}}{{b-a}}}{\int_{a}^{b}{f(x)\,dx}}\]

That is, there is a real number \(u\), \(a\leq u\leq b\), for which \[{\int_{a}^{b}{f(x)\,dx}}=f(u)\,(b-a) \]

EXAMPLE 7 Using the Mean Value Theorem for Integrals

Find the number(s) \(u\) guaranteed by the Mean Value Theorem forIntegrals for \(\int_{2}^{6}x^{2}dx.\)

Solution The Mean Value Theorem for Integrals states there is anumber \(u,\) \(2\leq u\leq 6,\) for which \[\int_{2}^{6}x^{2}dx=f(u) (6-2) =4u^{2}\qquad\qquad{\color{#0066A7}{\hbox{\(f(u)=u^2\)}}}\]

We integrate to obtain \[\int_{2}^{6}x^{2}dx=\left[\dfrac{x^{3}}{3}\right] _{2}^{6}=\dfrac{1}{3}(216-8) =\dfrac{208}{3}\]

Then\[\begin{eqnarray*}\begin{array}{rl@{\qquad}l@{\hskip-2pc}}\dfrac{208}{3} &= 4u^{2} \\u^{2} &= \dfrac{52}{3} & {\color{#0066A7}{2\leq u\leq 6}} \\u &= \sqrt{\dfrac{52}{3}}\approx 4.163 &{\color{#0066A7}{\hbox{Disregard the negative solutionsince \(u>0\).}}}\end{array}\end{eqnarray*}\]

NOW WORK

Problem 55.

5.7.3 Find the Average Value of a Function

We know from the Mean Value Theorem for Integrals that if a function \(f\) iscontinuous on a closed interval \([a,b]\), there is a real number \(u\), \(a\,\leq \,u\,\leq \,b\), for which \[\int_{a}^{b}f(x) \,dx = f(u) (b-a)\]

This means that if the function \(f\) is also nonnegative on the closedinterval \([a,b] ,\) the area enclosed by a rectangle of height \(f(u) \) and width \(b-a\) equals the area under the graph of \(f\)from \(a\) to \(b.\) See Figure 5.29.

Figure 5.29: \(\int^{b}_{a} f(x)\,dx = f(u)(b-a)\)

So if we replace \(f(x) \) on \([a,b] \) by \(f(u) ,\) weget a region with the same area. Consequently, \(f(u) \) can bethought of as an average value, or mean value,of \(f\) over \([a,b]\).

We can obtain the average value of \(f\) over \([a,b]\) for anyfunction \(f\) that is continuous on the closed interval \([a,b]\) by partitioning\([a,b]\) into \(n\) subintervals \[\lbrack a,x_{1}],  \lbrack x_{1},x_{2}],  \ldots ,  \lbrackx_{i-1},x_{i}],  \ldots ,  \lbrack x_{n-1},b] \]

374

each of length \(\Delta x=\dfrac{b-a}{n}\), and choosing a number \(u_{i}\) in each of the \(n\) subintervals. Then an approximation of the average value of \(f\) over the interval \([a,b]\) is \[\begin{equation*}\dfrac{{f(u_{1})+f(u_{2})+\cdots +f(u}_{n}{)}}{{n}}\tag{7}\end{equation*}\]

Now we multiply (7) by \(\dfrac{b-a}{b-a}\) to obtain \[\begin{array}{@{\hspace*{-2pc}}lcc}\dfrac{{f(u_{1})+f(u_{2})+\cdots +f(u_{n})}}{{n}} &=&{\dfrac{{1}}{{b-a}}}\left[ f(u_{1}){\dfrac{{b-a}}{{n}}}+f(u_{2}){\dfrac{{b-a}}{{n}}} \right.+ \left. \cdots +f(u_{n}){\dfrac{{b-a}}{{n}}}\right] \\&=&{\dfrac{{1}}{{b-a}}}[ f(u_{1})\,\Delta x+f(u_{2})\,\Deltax+\cdots +f(u_{n})\,\Delta x]\\& =& {\dfrac{{1}}{{b-a}}}{\sum\limits_{i=1}^{n}{f(u_{i})\,\Delta x}}\end{array}\]

This sum approximates the average value of \(f\). As the length ofeach subinterval gets smaller, the sums become better approximations to the average value of \(f\) on \([a,b] \).Furthermore, \({\sum\limits_{i=1}^{n}\,{f(u_{i})\,\Delta x}}\) are Riemannsums, so \(\lim\limits_{n\rightarrow \infty}\,{\sum\limits_{i=1}^{n}\,{f(u_{i})\,\Delta x}}\) is a definite integral. This suggests the followingdefinition:

IN WORDS

The average value \(\bar{y}=\dfrac{1}{b-a}\int_{a}^{b}{f(x)\,dx}\) of a function \(f\) equals thevalue \({f(u)}\) in the Mean Value Theorem for Integrals.

DEFINITION Average Value of a Function over an Interval

Let \(f\) be a function that is continuous on the closed interval \([a,b] .\) The average value \({\boldsymbol {\bar{y}}}\) of\({\boldsymbol f}\) over \(\boldsymbol{[a,b]}\) is\[\boxed{\bbox[5pt]{\bar{y}=\dfrac{1}{b-a} \int_{a}^{b}f(x)\,dx }}\tag{8}\]

EXAMPLE 8 Finding the Average Value of a Function

Find the average value of \(f(x)=3x-8\) on the closed interval \([0,2] .\)

Solution The average value of \(f(x)=3x-8\) on theclosed interval \([0,2]\) is given by \[\begin{eqnarray*}\bar{y} &=&\dfrac{1}{b-a}\int_{a}^{b}f(x)\,dx=\dfrac{1}{2-0}{\int_{0}^{2}}(3x-8) dx\\&=& \dfrac{1}{2}\left[\dfrac{3x^{2}}{2}-8x \right] _{0}^{2} =\dfrac{1}{2}(6-16) =-5\end{eqnarray*}\]

The average value of \(f\) on \([0,2] \) is \(\bar{y}=-5.\)

The function \(f\) and its average value are graphedin Figure 5.30.

Figure 5.30: \(f(x) = 3x-8,0 \leq x \leq 2\)

NOW WORK

Problem 61.