5.4: The Definite Integral
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In the preceding section we defined the area under a curve in terms of Riemann sums:
\[A=\lim_{n→∞}\sum_{i=1}^nf(x^∗_i)Δx.\]
However, this definition came with restrictions. We required \(f(x)\) to be continuous and nonnegative. Unfortunately, realworld problems don’t always meet these restrictions. In this section, we look at how to apply the concept of the area under the curve to a broader set of functions through the use of the definite integral.
Definition and Notation
The definite integral generalizes the concept of the area under a curve. We lift the requirements that \(f(x)\) be continuous and nonnegative, and define the definite integral as follows. Definition
Definition: definite integral
If \(f(x)\) is a function defined on an interval \([a,b],\) the definite integral of f from a to b is given by
\[∫^b_af(x)dx=\lim_{n→∞} \sum_{i=1}^nf(x^∗_i)Δx,\]
provided the limit exists. If this limit exists, the function \(f(x)\) is said to be integrable on [a,b], or is an integrable function.
The integral symbol in the previous definition should look familiar. We have seen similar notation in the chapter on Applications of Derivatives, where we used the indefinite integral symbol (without the a and b above and below) to represent an antiderivative. Although the notation for indefinite integrals may look similar to the notation for a definite integral, they are not the same. A definite integral is a number. An indefinite integral is a family of functions. Later in this chapter we examine how these concepts are related. However, close attention should always be paid to notation so we know whether we’re working with a definite integral or an indefinite integral.
Integral notation goes back to the late seventeenth century and is one of the contributions of Gottfried Wilhelm Leibniz, who is often considered to be the codiscoverer of calculus, along with Isaac Newton. The integration symbol ∫ is an elongated S, suggesting sigma or summation. On a definite integral, above and below the summation symbol are the boundaries of the interval, \([a,b].\) The numbers a and b are xvalues and are called the limits of integration; specifically, a is the lower limit and b is the upper limit. To clarify, we are using the word limit in two different ways in the context of the definite integral. First, we talk about the limit of a sum as \(n→∞.\) Second, the boundaries of the region are called the limits of integration.
We call the function \(f(x)\) the integrand, and the dx indicates that \(f(x)\) is a function with respect to x, called the variable of integration. Note that, like the index in a sum, the variable of integration is a dummy variable, and has no impact on the computation of the integral. We could use any variable we like as the variable of integration:
\[∫^b_af(x)dx=∫^b_af(t)dt=∫^b_af(u)du\]
Previously, we discussed the fact that if \(f(x)\) is continuous on \([a,b],\) then the limit \(\lim_{n→∞}\sum_{i=1}^nf(x^∗_i)Δx\) exists and is unique. This leads to the following theorem, which we state without proof.
Continuous Functions Are Integrable
If \(f(x)\) is continuous on \([a,b]\), then f is integrable on \([a,b].\)
Functions that are not continuous on \([a,b]\) may still be integrable, depending on the nature of the discontinuities. For example, functions with a finite number of jump discontinuities on a closed interval are integrable.
It is also worth noting here that we have retained the use of a regular partition in the Riemann sums. This restriction is not strictly necessary. Any partition can be used to form a Riemann sum. However, if a nonregular partition is used to define the definite integral, it is not sufficient to take the limit as the number of subintervals goes to infinity. Instead, we must take the limit as the width of the largest subinterval goes to zero. This introduces a little more complex notation in our limits and makes the calculations more difficult without really gaining much additional insight, so we stick with regular partitions for the Riemann sums.
Example \(\PageIndex{1}\): Evaluating an Integral Using the Definition
Note to Calc I students: this example is more detailed than the exercises you need to know how to do.
Use the definition of the definite integral to evaluate \(∫^2_0x^2dx.\) Use a rightendpoint approximation to generate the Riemann sum.
Solution
We first want to set up a Riemann sum. Based on the limits of integration, we have \(a=0\) and \(b=2\). For \(i=0,1,2,…,n\), let \(P={x_i}\) be a regular partition of \([0,2].\) Then
\[Δx=\dfrac{b−a}{n}=\dfrac{2}{n}.\]
Since we are using a rightendpoint approximation to generate Riemann sums, for each i, we need to calculate the function value at the right endpoint of the interval \([x_{i−1},x_i].\) The right endpoint of the interval is \(x_i\), and since P is a regular partition,
\[x_i=x_0+iΔx=0+i[\dfrac{2}{n}]=\dfrac{2i}{n}.\]
Thus, the function value at the right endpoint of the interval is
\[f(x_i)=x^2_i=(\dfrac{2i}{n})^2=\dfrac{4i^2}{n^2}.\]
Then the Riemann sum takes the form
\(\sum_{i=1}^nf(x_i)Δx=\sum_{i=1}^n(\dfrac{4i^2}{n^2})\dfrac{2}{n}=\sum_{i=1}^n\dfrac{8i^2}{n^3}=\dfrac{8}{n^3}\sum_{i=1}^ni^2\).
Using the summation formula for \(\sum_{i=1}^ni^2\), we have
\(\sum_{i=1}^nf(x_i)Δx=\dfrac{8}{n^3}\sum_{i=1}^ni^2\)
\(=\dfrac{8}{n^3}[\dfrac{n(n+1)(2n+1)}{6}]\)
\(=\dfrac{8}{n^3}[\dfrac{2n^3+3n^2+n}{6}]\)
\(=\dfrac{16n^3+24n^2+8n}{6n^3}\)
\(=\dfrac{8}{3}+\dfrac{4}{n}+\dfrac{4}{3n^2}\).
Now, to calculate the definite integral, we need to take the limit as \(n→∞\). We get
\(∫^2_0x^2dx=\lim_{n→∞}\sum_{i=1}^nf(x_i)Δx=\lim_{n→∞}(\dfrac{8}{3}+\dfrac{4}{n}+\dfrac{4}{3n^2})\)
\(=\lim_{n→∞}(\dfrac{8}{3})+\lim_{n→∞}(\dfrac{4}{n})+\lim_{n→∞}(\dfrac{4}{3n^2})=\dfrac{8}{3}+0+0=\dfrac{8}{3}\).
Exercise \(\PageIndex{1}\)
Use the definition of the definite integral to evaluate \(∫^3_0(2x−1)dx\).
Use a rightendpoint approximation to generate the Riemann sum.
 Hint

Use the solving strategy from Example \(\PageIndex{1}\).
 Answer

\(6\)
Evaluating Definite Integrals
Evaluating definite integrals this way can be quite tedious because of the complexity of the calculations. Later in this chapter we develop techniques for evaluating definite integrals without taking limits of Riemann sums. However, for now, we can rely on the fact that definite integrals represent the area under the curve, and we can evaluate definite integrals by using geometric formulas to calculate that area. We do this to confirm that definite integrals do, indeed, represent areas, so we can then discuss what to do in the case of a curve of a function dropping below the xaxis.
Example \(\PageIndex{2}\): Using Geometric Formulas to Calculate Definite Integrals
Use the formula for the area of a circle to evaluate \(∫^6_3\sqrt{9−(x−3)^2}\,dx\).
Solution
The function describes a semicircle with radius 3. To find
\[∫^6_3\sqrt{9−(x−3)^2}\,dx\]
we want to find the area under the curve over the interval \([3,6].\) The formula for the area of a circle is \(A=πr^2\). The area of a semicircle is just onehalf the area of a circle, or \(A=(\dfrac{1}{2})πr^2\). The shaded area in Figure covers onehalf of the semicircle, or \(A=(\dfrac{1}{4})πr^2\). Thus,
\[∫^6_3\sqrt{9−(x−3)^2}=\dfrac{1}{4}π(3)^2=\dfrac{9}{4}π≈7.069.\]
Figure \(\PageIndex{1}\): The value of the integral of the function \(f(x)\) over the interval \([3,6]\) is the area of the shaded region.
Exercise \(\PageIndex{2}\)
Use the formula for the area of a trapezoid to evaluate \(∫^4_2(2x+3)dx\).
 Hint

Graph the function \(f(x)\) and calculate the area under the function on the interval \([2,4].\)
 Answer

18 square units
Area and the Definite Integral
When we defined the definite integral, we lifted the requirement that \(f(x)\) be nonnegative. But how do we interpret “the area under the curve” when \(f(x)\) is negative?
Net Signed Area
Let us return to the Riemann sum. Consider, for example, the function \(f(x)=2−2x^2\) (shown in Figure) on the interval \([0,2]\). Use \(n=8\) and choose {\(x^∗_i\)} as the left endpoint of each interval. Construct a rectangle on each subinterval of height \(f(x^∗_i)\) and width \(Δx\). When \(f(x^∗_i)\) is positive, the product \(f(x^∗_i)Δx\) represents the area of the rectangle, as before. When \(f(x^∗_i)\) is negative, however, the product \(f(x^∗_i)Δx\) represents the negative of the area of the rectangle. The Riemann sum then becomes
\[\sum_{i=1}^8f(x^∗_i)Δx=\]
(Area of rectangles above thexaxis)−(Area of rectangles below thexaxis)
Figure \(\PageIndex{2}\): For a function that is partly negative, the Riemann sum is the area of the rectangles above the xaxis less the area of the rectangles below the xaxis.
Taking the limit as \(n→∞,\) the Riemann sum approaches the area between the curve above the xaxis and the xaxis, less the area between the curve below the xaxis and the xaxis, as shown in Figure. Then,
\[\int^2_0f(x)dx=\lim_{n→∞}\sum_{i=1}^nf(c_i)Δx=A_1−A_2.\]
The quantity \(A_1−A_2\) is called the net signed area.
Figure \(\PageIndex{3}\): In the limit, the definite integral equals area A1 less area A2, or the net signed area.
Notice that net signed area can be positive, negative, or zero. If the area above the xaxis is larger, the net signed area is positive. If the area below the xaxis is larger, the net signed area is negative. If the areas above and below the xaxis are equal, the net signed area is zero.
Example \(\PageIndex{3}\): Finding the Net Signed Area
Find the net signed area between the curve of the function \(f(x)=2x\) and the xaxis over the interval \([−3,3].\)
Solution
The function produces a straight line that forms two triangles: one from \(x=−3\) to \(x=0\) and the other from \(x=0\) to \(x=3\) (Figure). Using the geometric formula for the area of a triangle, \(A=\dfrac{1}{2}bh\), the area of triangle A1, above the axis, is
\(A_1=\dfrac{1}{2}3(6)=9\),
where 3 is the base and \(2(3)=6\) is the height. The area of triangle A2, below the axis, is
\(A_2=\dfrac{1}{2}(3)(6)=9,\)
where 3 is the base and 6 is the height. Thus, the net area is
\(∫^3_{−3}2xdx=A_1−A_2=9−9=0.\)
Figure \(\PageIndex{4}\): The area above the curve and below the xaxis equals the area below the curve and above the xaxis.
Analysis
If A1 is the area above the xaxis and A2 is the area below the xaxis, then the net area is \(A_1−A_2\). Since the areas of the two triangles are equal, the net area is zero.
Exercise \(\PageIndex{3}\)
Find the net signed area of \(f(x)=x−2\) over the interval \([0,6]\), illustrated in the following image.
 Hint

Use the solving method described in Example \(\PageIndex{3}\).
 Answer

6
Total Area
One application of the definite integral is finding displacement when given a velocity function. If \(v(t)\) represents the velocity of an object as a function of time, then the area under the curve tells us how far the object is from its original position. This is a very important application of the definite integral, and we examine it in more detail later in the chapter. For now, we’re just going to look at some basics to get a feel for how this works by studying constant velocities.
When velocity is a constant, the area under the curve is just velocity times time. This idea is already very familiar. If a car travels away from its starting position in a straight line at a speed of 75 mph for 2 hours, then it is 150 mi away from its original position (Figure). Using integral notation, we have
\(∫^2_075dt=150\).
Figure \(\PageIndex{5}\): The area under the curve \(v(t)=75\) tells us how far the car is from its starting point at a given time.
In the context of displacement, net signed area allows us to take direction into account. If a car travels straight north at a speed of 60 mph for 2 hours, it is 120 mi north of its starting position. If the car then turns around and travels south at a speed of 40 mph for 3 hours, it will be back at it starting position (Figure). Again, using integral notation, we have
\[\int^2_060\,dt+∫^5_2−40\,dt=120−120=0.\]
In this case the displacement is zero.
Figure \(\PageIndex{6}\): The area above the axis and the area below the axis are equal, so the net signed area is zero.
Suppose we want to know how far the car travels overall, regardless of direction. In this case, we want to know the area between the curve and the xaxis, regardless of whether that area is above or below the axis. This is called the total area.
Graphically, it is easiest to think of calculating total area by adding the areas above the axis and the areas below the axis (rather than subtracting the areas below the axis, as we did with net signed area). To accomplish this mathematically, we use the absolute value function. Thus, the total distance traveled by the car is
\[\int^2_060dt+∫^5_2−40dt=∫^2_060dt+∫^5_240dt=120+120=240.\]
Bringing these ideas together formally, we state the following definitions.
Definition: net signed area
Let \(f(x)\) be an integrable function defined on an interval \([a,b]\). Let \(A_1\) represent the area between \(f(x)\) and the xaxis that lies above the axis and let \(A_2\) represent the area between \(f(x)\) and the xaxis that lies below the axis. Then, the net signed area between \(f(x)\) and the xaxis is given by
\[∫^b_af(x)dx=A_1−A_2.\]
The total area between \(f(x)\) and the xaxis is given by
\[∫^b_af(x)dx=A_1+A_2.\]
Example \(\PageIndex{4}\): Finding the Total Area
Find the total area between \(f(x)=x−2\) and the xaxis over the interval \([0,6].\)
Solution
Calculate the xintercept as \((2,0)\) (set \(y=0,\) solve for x). To find the total area, take the area below the xaxis over the subinterval \([0,2]\) and add it to the area above the xaxis on the subinterval \([2,6]\) (Figure).
Figure \(\PageIndex{7}\): The total area between the line and the xaxis over \([0,6]\) is \(A_2\) plus \(A_1\).
We have
\(∫^6_0(x−2)dx=A_2+A_1.\)
Then, using the formula for the area of a triangle, we obtain
\(A_2=\dfrac{1}{2}bh=\dfrac{1}{2}⋅2⋅2=2\)
\(A_1=\dfrac{1}{2}bh=\dfrac{1}{2}⋅4⋅4=8\).
The total area, then, is
\(A_1+A_2=8+2=10\).
Exercise \(\PageIndex{4}\)
Find the total area between the function \(f(x)=2x\) and the xaxis over the interval \([−3,3].\)
 Hint

Review the solving strategy in Example \(\PageIndex{4}\).
 Answer

\(18\)
Properties of the Definite Integral
The properties of indefinite integrals apply to definite integrals as well. Definite integrals also have properties that relate to the limits of integration. These properties, along with the rules of integration that we examine later in this chapter, help us manipulate expressions to evaluate definite integrals.
Rule: Properties of the Definite Integral
1.
\[∫^a_af(x)\,dx=0\]
If the limits of integration are the same, the integral is just a line and contains no area.
2.
\[∫^a_bf(x)\,dx=−∫^b_af(x)dx\]
If the limits are reversed, then place a negative sign in front of the integral.
3.
\[∫^b_a[f(x)+g(x)]\,dx=∫^b_af(x)dx+∫^b_ag(x)dx\]
The integral of a sum is the sum of the integrals.
4.
\[∫^b_a[f(x)−g(x)]\,dx=∫^b_af(x)dx−∫^b_ag(x)dx\]
The integral of a difference is the difference of the integrals
5.
\[∫^b_acf(x)\,dx=c∫^b_af(x)\]
for constant c. The integral of the product of a constant and a function is equal to the constant multiplied by the integral of the function.
6.
\[∫^b_af(x)\,dx=∫^c_af(x)\,dx+∫^b_cf(x)\,dx\]
Although this formula normally applies when c is between a and b, the formula holds for all values of \(a\), \(b\), and \(c\), provided \(f(x)\) is integrable on the largest interval.
Example \(\PageIndex{5}\): Using the Properties of the Definite Integral
Use the properties of the definite integral to express the definite integral of \(f(x)=−3x^3+2x+2\) over the interval \([−2,1]\) as the sum of three definite integrals.
Solution
Using integral notation, we have \(∫^1_{−2}(−3x^3+2x+2)dx.\) We apply properties 3. and 5. to get
\(∫^1_{−2}(−3x^3+2x+2)dx=∫^1_{−2}−3x^3dx+∫^1_{−2}2xdx+∫^1_{−2}2dx\)
\(=−3∫^1_{−2}x^3dx+2∫^1_{−2}xdx+∫^1_{−2}2dx.\)
Exercise \(\PageIndex{5}\)
Use the properties of the definite integral to express the definite integral of \(f(x)=6x^3−4x^2+2x−3\) over the interval \([1,3]\) as the sum of four definite integrals.
 Hint

Use the solving strategy from Example \(\PageIndex{5}\) and the properties of definite integrals.
 Answer

\[6∫^3_1x^3dx−4∫^3_1x^2dx+2∫^3_1xdx−∫^3_13dx\]
Example \(\PageIndex{6}\): Using the Properties of the Definite Integral
If it is known that \(∫^8_0f(x)dx=10\) and \(∫^5_0f(x)dx=5\), find the value of \(∫^8_5f(x)dx\).
Solution
By property 6.,
\(∫^b_af(x)dx=∫^c_af(x)dx+∫^b_cf(x)dx.\)
Thus,
\(∫^8_0f(x)dx=∫^5_0f(x)dx+∫^8_5f(x)dx\)
\(10=5+∫^8_5f(x)dx\)
\(5=∫^8_5f(x)dx\).
Exercise \(\PageIndex{6}\)
If it is known that \(∫^5_1f(x)dx=−3\) and \(∫^5_2f(x)dx=4\), find the value of \(∫^2_1f(x)dx.\)
 Hint

Use the solving strategy from Example \(\PageIndex{6}\) and the rule on properties of definite integrals.
 Answer

\(−7\)
Comparison Properties of Integrals
A picture can sometimes tell us more about a function than the results of computations. Comparing functions by their graphs as well as by their algebraic expressions can often give new insight into the process of integration. Intuitively, we might say that if a function \(f(x)\) is above another function \(g(x)\), then the area between \(f(x)\) and the xaxis is greater than the area between \(g(x)\) and the xaxis. This is true depending on the interval over which the comparison is made. The properties of definite integrals are valid whether \(a<b,a=b\), or \(a>b\). The following properties, however, concern only the case \(a≤b\), and are used when we want to compare the sizes of integrals.
Comparison Theorem
i. If \(f(x)≥0\) for \(a≤x≤b\), then
\[∫^b_af(x)dx≥0.\]
ii. If \(f(x)≥g(x)\) for \(a≤x≤b\), then
\[∫^b_af(x)dx≥∫^b_ag(x)dx.\]
iii. If m and M are constants such that \(m≤f(x)≤M\) for \(a≤x≤b\), then
\[m(b−a)≤∫^b_af(x)dx≤M(b−a).\]
Example \(\PageIndex{7}\): Comparing Two Functions over a Given Interval
Compare \(f(x)=\sqrt{1+x^2}\) and \(g(x)=\sqrt{1+x}\) over the interval \([0,1]\).
Solution
Graphing these functions is necessary to understand how they compare over the interval \([0,1].\) Initially, when graphed on a graphing calculator, \(f(x)\) appears to be above \(g(x)\) everywhere. However, on the interval \([0,1]\), the graphs appear to be on top of each other. We need to zoom in to see that, on the interval \([0,1],g(x)\) is above \(f(x)\). The two functions intersect at \(x=0\) and \(x=1\) (Figure).
Figure \(\PageIndex{8}\): (a) The function \(f(x)\) appears above the function \(g(x)\) except over the interval \([0,1]\) (b) Viewing the same graph with a greater zoom shows this more clearly.
We can see from the graph that over the interval \([0,1],g(x)≥f(x)\). Comparing the integrals over the specified interval \([0,1],\) we also see that \(∫^1_0g(x)dx≥∫^1_0f(x)dx\) (Figure). The thin, redshaded area shows just how much difference there is between these two integrals over the interval \([0,1].\)
Figure \(\PageIndex{9}\): (a) The graph shows that over the interval \([0,1],g(x)≥f(x),\) where equality holds only at the endpoints of the interval. (b) Viewing the same graph with a greater zoom shows this more clearly.
Key Concepts
 The definite integral can be used to calculate net signed area, which is the area above the xaxis less the area below the xaxis. Net signed area can be positive, negative, or zero.
 The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.
 Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.
 The properties of definite integrals can be used to evaluate integrals.
 The area under the curve of many functions can be calculated using geometric formulas.
Key Equations
 Definite Integral
\(\displaystyle∫^b_af(x)dx=\lim_{n→∞}\sum_{i=1}^nf(x^∗_i)Δx\)
 Properties of the Definite Integral
\(\displaystyle∫^a_af(x)dx=0\)
\(\displaystyle∫^a_bf(x)dx=−∫^b_af(x)dx\)
\(\displaystyle∫^b_a[f(x)+g(x)]dx=∫^b_af(x)dx+∫^b_ag(x)dx\)
\(\displaystyle∫^b_a[f(x)−g(x)]dx=∫^b_af(x)dx−∫^b_ag(x)dx\)
\(\displaystyle∫^b_acf(x)dx=c∫^b_af(x)\) for constant c
\(\displaystyle∫^b_af(x)dx=∫^c_af(x)dx+∫^b_cf(x)dx\)
Glossary
 definite integral
 a primary operation of calculus; the area between the curve and the xaxis over a given interval is a definite integral
 integrable function
 a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as n goes to infinity exists
 integrand
 the function to the right of the integration symbol; the integrand includes the function being integrated
 limits of integration
 these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated
 net signed area
 the area between a function and the xaxis such that the area below the xaxis is subtracted from the area above the xaxis; the result is the same as the definite integral of the function
 total area
 total area between a function and the xaxis is calculated by adding the area above the xaxis and the area below the xaxis; the result is the same as the definite integral of the absolute value of the function
 variable of integration
 indicates which variable you are integrating with respect to; if it is x, then the function in the integrand is followed by dx
Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CCBYSANC 4.0 license. Download for free at http://cnx.org.