Chapter 5. The Integral

Managing the Klamath River

The Klamath River starts in the eastern lava plateaus of Oregon, passes through a farming region of that state, and then crosses northern California. Because it is the largest river in the region, and runs through a variety of terrain and has different land uses, it is important in a number of ways. Historically, the Klamath supported large salmon runs. It is used to irrigate agricultural lands, and to generate electric power for the region. Downstream, it runs through federal wild lands, and is used for fishing, rafting, and kayaking.

If a river is to be well-managed for such a variety of uses, its flow must be understood. For that reason, the U.S. Geological Survey (USGS) maintains a number of gauges along the Klamath. These typically measure the depth of the river, which can then be expressed as a rate of flow in cubic feet per second (\(\text{ft}^{3}/\text{s}\)). In order to understand the river flow fully, one must be able to find the total amount of water that flows down the river over any period of time.

OBJECTIVES

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

1. Approximate the area under the graph of a function (p. 2)
2. Find the area under the graph of a function (p. 6)

EXAMPLE 1 Approximating the Area Under the Graph of a Function

Approximate the area \(A\) enclosed by the graph of \(f(x) =\dfrac{1}{2}x+3\), the \(x\)-axis, and the lines \(x=2\) and \(x=4\).

SolutionFigure 5.4 illustrates the area \(A\) to be approximated.

We begin by drawing a rectangle of base \(4-2=2\) and height \(f(2)=4\). The area of the rectangle, \(2\cdot 4=8\), approximates the area \(A\), but it underestimates \(A\), as seen in Figure 5.5(a).

Alternatively, \(A\) can be approximated by a rectangle of base \(4-2=2\) and height \(f (4) =5\). See Figure 5.5(b). This approximation of the area equals \(2\cdot 5=10\), but it overestimates \(A\). We conclude that \[ 8 \lt A \lt 10 \]

The approximation of the area \(A\) can be improved by dividing the closed interval \([2, 4]\) into two subintervals, \([2, 3]\) and \([3, 4]\). Now we draw two rectangles: one rectangle with base \(3-2=1\) and height \(f(2) =\dfrac{1}{2}\cdot 2+3=4\); the other rectangle with base \(4-3=1\) and height \(f (3) =\dfrac{1}{2}\cdot 3+3=\dfrac{9}{2}\). As Figure 5.6(a) illustrates, the sum of the areas of the two rectangles \[ 1\cdot 4+1\cdot \dfrac{9}{2}=\dfrac{17}{2}=8.5 \] underestimates the area.

Now we repeat this process by drawing two rectangles, one of base \(1\) and height \(f(3) =\dfrac{9}{2}\); the other of base \(1\) and height \(f(4) =\dfrac{1}{2}\cdot 4+3=5\). As Figure 5.6(b) illustrates, the sum of the areas of these two rectangles, \[ 1\cdot \dfrac{9}{2}+1\cdot 5=\dfrac{19}{2}=9.5 \]

overestimates the area. We conclude that \[ 8.5 \lt A \lt 9.5 \]

obtaining a better approximation to the area.

NOW WORK

Problem 5

NEED TO REVIEW?

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

NEED TO REVIEW?

Summation notation is discussed in Appendix A.5, pp. A-38-A-43.

EXAMPLE 2 Approximating Area Using Lower Sums

Approximate the area \(A\) under the graph of \(f(x) = x^{2}\) from \(0\) to \(10\) by using lower sums \(s_{n}\) (rectangles that lie under the graph) for:

1. \(n = 2\) subintervals
2. \(n = 5\) subintervals
3. \(n = 10\) subintervals

Solution(a) For \(n=2\), we partition the closed interval \([0,10]\) into two subintervals \([0,5]\) and \([5,10]\), each of length \(\Delta x=\dfrac{10-0}{2}=5\). See Figure 5.9(a). To compute \(s_{2}\), we need to know where \(f\) attains its minimum value in each subinterval. Since \(f\) is an increasing function, the absolute minimum is attained at the left endpoint of each subinterval. So, for \(n=2\), the minimum of \(f\) on \([0,5]\) occurs at \(0\) and the minimum of \(f\) on \([5,10]\) occurs at 5. The lower sum \(s_{2}\) is

(b) For \(n=5\), partition the interval \([0,10]\) into five subintervals \([0,2]\), \([2,4]\), \([4,6]\), \([6,8]\), \([8,10]\), each of length \(\Delta x = \dfrac{10-0}{5}=2\). See Figure 5.9(b). The lower sum \(s_{5}\) is \[\begin{eqnarray*} s_{5} &=&\sum\limits_{i=1}^{5} f(c_{i})\Delta x=\Delta x\sum\limits_{i=1}^{5}f(c_{i})=2 [ f(0)+f(2)+f(4) +f(6)+f(8)] \\ &=&2( 0+4+16+36+64 ) =240 \end{eqnarray*}\]

(c) For \(n=10\), partition \([0,10]\) into \(10\) subintervals, each of length \(\Delta x=\dfrac{10-0}{10}=1\). See Figure 5.9(c). The lower sum \(s_{10}\) is \[ \begin{eqnarray*} s_{10} &=&\sum\limits_{i=1}^{10} f(c_{i})\Delta x=\Delta x\sum\limits_{i=1}^{10}f(c_{i})=1[ f(0)+f(1)+f(2)+\cdots +f(9) ] \\ &=&0+1+4+9+16+25+36+49+64+81=285 \end{eqnarray*}\]

NOW WORK

Problem 13(a).

DEFINITION Area \({A}\) Under the Graph of a Function from \({a}\) to \({b}\)

Suppose a function \(f\) is nonnegative and continuous on a closed interval \([a,b]\). Partition \([a,b]\) into \(n\) subintervals \([x_{0}, x_{1}], [x_{1}, x_{2}], \ldots, [x_{i-1}, x_{i}], \ldots, [x_{n-1}, x_{n}]\), each of length \[ \Delta x=\dfrac{b-a}{n} \]

In each subinterval \([ x_{i-1},x_{i}]\), let \(f (c_{i})\) equal the absolute minimum value of \(f\) on this subinterval. Form the lower sums \[ s_{n}=\sum\limits_{i= 1}^{n}f(c_{i})\Delta x=f(c_{\mathbf{1}})\Delta x+ \cdots +f(c_{n})\Delta x \]

The area \(\boldsymbol{A}\) under the graph of \(\boldsymbol{f}\) from \(\boldsymbol{a}\) to \(\boldsymbol{b}\) is the number \[ \boxed{\bbox[5pt] {A=\lim\limits_{n\rightarrow \infty }s_{n} }} \]

EXAMPLE 3 Finding Area Using Upper Sums

Find the area \(A\) under the graph of \(f(x) = 3x\) from \(0\) to \(10\) using upper sums \(S_{n}\) (rectangles that lie above the graph of \(f)\). Then \(A = \lim\limits_{n\rightarrow \infty }S_{n}\).

SolutionFigure 5.12 illustrates the area \(A\). We partition the closed interval \([0,10]\) into \(n\) subintervals \[ {[{x_{0},x_{1}}]},\quad {[{x_{1},x_{2}}]}, \ldots, [x_{i-1},x_{i}] , \ldots ,\quad {{[{x_{n-1},x_{n}}]}} \]

Figure 5.12: \(f(x)=3x, 0 \leq x \leq 10\).

where \[ 0=x_{0}\lt x_{1}\lt x_{2}\lt\cdots \lt x_{i}\lt\cdots \lt x_{n-1}\lt x_{n}=10 \]

and each subinterval is of length \[ \Delta x = {\dfrac{{10 - 0}}{{n}}} = {\dfrac{{10}}{{n}}} \]

The coordinates of the endpoints of each subinterval, written in terms of \(n\), are \[ \begin{eqnarray*} x_0 &=& 0,x_1 = \frac{{10}} {n},x_2 = 2\left( {\frac{{10}} {n}} \right), \ldots ,x_{i - 1} = (i - 1)\left( {\frac{{10}} {n}} \right), \\ x_i & =& i\left( {\frac{{10}} {n}} \right), \ldots ,x_n = n\left( {\frac{{10}} {n}} \right) = 10 \\ \end{eqnarray*} \]

as illustrated in Figure 5.13.

To find \(A\) using upper sums \(S_{n}\) (rectangles that lie above the graph of \(f\)), we need the absolute maximum value of \(f\) in each subinterval. Since \(f(x)=3x\) is an increasing function, the absolute maximum occurs at the right endpoint \(x_{i}= i\!\left(\dfrac{10}{n}\right)\) of each subinterval. So, \[ \begin{eqnarray*} &S_{n}=\sum\limits_{i=1}^{n} f (C_i)&\Delta x = \sum \limits_{i=1}^{n}3x_{i}\cdot&\dfrac{10}{n} = {\sum\limits_{i=1}^{n}}\left( 3\cdot \dfrac{10i}{n}\right) \left(\dfrac{10}{n}\right) = \sum\limits_{i=1}^{n}\dfrac{300}{n^2}i \nonumber\\ &&\underset{\hbox{\(\color{#0066A7}{\Delta x = \dfrac{10}{n}}\)}}{\color{#0066A7}{\uparrow}} &\underset{\hbox{\(\color{#0066A7}{ {x}_{i} = \dfrac{10i}{n}}\)}}{\color{#0066A7}{\uparrow}} \end{eqnarray*} \]

Using summation properties, we get \[ S_{n}= \sum\limits_{i=1}^{n} \dfrac{300}{n^2} i =\dfrac{300}{n^2} \sum\limits_{i=1}^{n} i = \dfrac{300}{n^2}\ \dfrac{n(n+1)}{2} = 150\left(\dfrac{n+1}{n}\right) =150\left(1+\dfrac{1}{n}\right) \]

Then \[ A=\lim\limits_{n\rightarrow \infty }S_{n}=\lim\limits_{n\rightarrow \infty } \left[150 \left( 1+\dfrac{1}{n} \right) \right] =150\lim\limits_{n\rightarrow \infty }\left(1+\dfrac{1}{n}\right) =150 \]

The area \(A\) under the graph of \(f(x)=3x\) from \(0\) to \(10\) is \(150\) square units.

NOTE

The area found in Example 3 is that of a triangle. So, we can verify that \(A=150\) by using the formula for the area \(A\) of a triangle with base \(b\) and height \(h\): \[ A=\dfrac{1}{2}bh = \dfrac{1}{2} (10)(30) = 150 \].

EXAMPLE 4 Finding Area Using Lower Sums

Find the area \(A\) under the graph of \(f(x) = 16 - x^{2}\) from \(0\) to \(4\) by using lower sums \(s_{n}\) (rectangles that lie below the graph of \(f\)). Then \(A=\lim\limits_{n\rightarrow \infty}{}s_{n}\).

SolutionFigure 5.14 shows the area under the graph of \(f\) and a typical rectangle that lies below the graph. We partition the closed interval \([0,4]\) into \(n\) subintervals \[ {[{x_{0},x_{1}}]},\quad {[{x_{1},x_{2}}]}, \ldots , [x_{i-1},x_{i}], \ldots , {[{x_{n-1},x_{n}}]} \]

Figure 5.14: \(f(x)=16-x^{2},0 \leq x \leq 4\).

where \[ 0=x_{0}\lt x_{1}\lt \cdots \lt x_{i}\lt \cdots \lt x_{n-1} \lt x_{n}=4 \]

and each interval is of length \[ \Delta x = \dfrac{{4-0}}{n} = \dfrac{4}{n} \]

As Figure 5.15 illustrates, the endpoints of each subinterval, written in terms of \(n\), are \[ \begin{array}{l} {x_0 = 0,x_1 = 1\left( {\frac{4} {n}} \right),x_2 = 2\left( {\frac{4} {n}} \right), \cdots ,x_{i - 1} = (i - 1)\left( {\frac{4} {n}} \right),} \\ {x_i = i\left( {\frac{4} {n}} \right), \cdots ,x_n = n\left( {\frac{4} {n}} \right) = 4} \\ \end{array} \]

To find \(A\) using lower sums \(s_{n}\) (rectangles that lie below the graph of \(f\)), we must find the absolute minimum value of \(f\) on each subinterval. Since the function \(f\) is a decreasing function, the absolute minimum occurs at the right endpoint of each subinterval. So, \[ s_{n}={\sum\limits_{i=1}^{n}} f (c_{i})\Delta x \]

Since \(c_{i}=i\!\left( \dfrac{4}{n}\right) =\dfrac{4i}{n}\) and \(\Delta x=\dfrac{4}{n}\), we have \[ \begin{array}{l} {s_n } & { = \sum\limits_{i = 1}^n {f(c_i )\Delta x = \sum\limits_{i = 1}^n {\left[ {16 - \left( {\frac{{4i}} {n}} \right)^2 } \right]} \underset{\color{#0066A7}{f(c_i ) = 16 - c_i^2 ;c_i = \frac{{4i}}{4}}}{\left( {\frac{4} {n}} \right)} }} \\ {} & { = \sum\limits_{i = 1}^n {\left[ {\frac{{64}} {n} - \frac{{64i^2 }} {{n^3 }}} \right]} } \\ {} & { = \frac{{64}} {n}\sum\limits_{i = 1}^n {1 - \frac{{64}} {{n^3 }}\sum\limits_{i = 1}^n {i^2 } } } \\ {} & { = \frac{{64}} {n}(n) - \frac{{64}} {{n^3 }}\left[ {\frac{{n(n + 1)(2n + 1)}} {6}} \right] = 64 - \frac{{32}} {{3n^2 }}[2n^2 + 3n + 1]} \\ {} & { = 64 - \frac{{64}} {3} - \frac{{32}} {n} - \frac{{32}} {{3n^2 }} = \frac{{128}} {3} - \frac{{32}} {n} - \frac{{32}} {{3n^2 }}} \\ \end{array} \]

Then \[ A=\lim\limits_{n\rightarrow \infty }s_{n}={\lim\limits_{n\rightarrow \infty } }\left( {{\dfrac{{128}}{{3}}}}-{{\dfrac{{32}}{{n}}}-{\dfrac{{32}}{{3n^{2}}}}} \right) ={\dfrac{{128}}{{3}}} \]

The area \(A\) under the graph of \(f (x) =16-x^{2}\) from \(0\) to \(4\) is \(\dfrac{128}{3}\) square units.

NOW WORK

Problem 29

OBJECTIVES

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

1. Define a definite integral as the limit of Riemann sums (p. 11)
2. Find a definite integral using the limit of Riemann sums (p. 14)

THEOREM The Integral of the Sum of Two Functions

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

CHAPTER 5 PROJECT Managing the Klamath River

There is a gauge on the Klamath River, just downstream from the dam at Keno, Oregon. The U.S. Geological Survey has posted flow rates for this gauge every month since 1930. The averages of these monthly measurements since 1930 are given in Table 2. Notice that the data in Table 2 measure the rate of change of the volume \(V\) in cubic feet of water each second over one year; that is, the table gives \(\dfrac{dV}{dt}=V^\prime (t)\) in \(\hbox{cubic feet per second}\), where \(t\) is in months.