Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

5.2: The Definite Integral

( \newcommand{\kernel}{\mathrm{null}\,}\)

Learning Objectives
  • State the definition of the definite integral.
  • Explain the terms integrand, limits of integration, and variable of integration.
  • Explain when a function is integrable.
  • Describe the relationship between the definite integral and net area.
  • Use geometry and the properties of definite integrals to evaluate them.
  • Calculate the average value of a function.

In the preceding section we defined the area under a curve in terms of Riemann sums:

๐ด=lim๐‘›โ†’โˆžโก๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘ฅโˆ—๐‘–)โขฮ”โข๐‘ฅ.

However, this definition came with restrictions. We required ๐‘“โก(๐‘ฅ) to be continuous and nonnegative. Unfortunately, real-world problems donโ€™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 ๐‘“โก(๐‘ฅ) be continuous and nonnegative, and define the definite integral as follows.

Definition: Definite Integral

If ๐‘“โก(๐‘ฅ) is a function defined on an interval [๐‘Ž,๐‘], the definite integral of ๐‘“ from ๐‘Ž to ๐‘ is given by

โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ=lim๐‘›โ†’โˆžโก๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘ฅโˆ—๐‘–)โขฮ”โข๐‘ฅ,

provided the limit exists. If this limit exists, the function ๐‘“โก(๐‘ฅ) is said to be integrable on [๐‘Ž,๐‘], 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 ๐‘Ž and ๐‘ 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 weโ€™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 โˆซ is an elongated ๐‘†, suggesting sigma or summation. On a definite integral, above and below the summation symbol are the boundaries of the interval, [๐‘Ž,๐‘]. The numbers ๐‘Ž and ๐‘ are ๐‘ฅ-values and are called the limits of integration; specifically, ๐‘Ž is the lower limit and ๐‘ 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 ๐‘› โ†’โˆž. Second, the boundaries of the region are called the limits of integration.

We call the function ๐‘“โก(๐‘ฅ) the integrand, and the ๐‘‘โข๐‘ฅ indicates that ๐‘“โก(๐‘ฅ) is a function with respect to ๐‘ฅ, 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:

โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ=โˆซ๐‘๐‘Ž๐‘“โก(๐‘ก)๐‘‘๐‘ก=โˆซ๐‘๐‘Ž๐‘“โก(๐‘ข)๐‘‘๐‘ข

Previously, we discussed the fact that if ๐‘“โก(๐‘ฅ) is continuous on [๐‘Ž,๐‘], then the limit lim๐‘›โ†’โˆžโก๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘ฅโˆ—๐‘–)โขฮ”โข๐‘ฅ exists and is unique. This leads to the following theorem, which we state without proof.

Continuous Functions Are Integrable

If ๐‘“โก(๐‘ฅ) is continuous on [๐‘Ž,๐‘], then ๐‘“ is integrable on [๐‘Ž,๐‘].

Functions that are not continuous on [๐‘Ž,๐‘] may still be integrable, depending on the nature of the discontinuities. For example, functions with a finite number of jump discontinuities or removable 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 5.2.1: Evaluating an Integral Using the Definition

Use the definition of the definite integral to evaluate โˆซ20๐‘ฅ2 ๐‘‘๐‘ฅ. Use a right-endpoint approximation to generate the Riemann sum.

Solution

We first want to set up a Riemann sum. Based on the limits of integration, we have ๐‘Ž =0 and ๐‘ =2. For ๐‘– =0,1,2,โ€ฆ,๐‘›, let ๐‘ƒ =๐‘ฅ๐‘– be a regular partition of [0,2]. Then

ฮ”โข๐‘ฅ=๐‘โˆ’๐‘Ž๐‘›=2๐‘›.

Since we are using a right-endpoint approximation to generate Riemann sums, for each ๐‘–, we need to calculate the function value at the right endpoint of the interval [๐‘ฅ๐‘–โˆ’1,๐‘ฅ๐‘–]. The right endpoint of the interval is ๐‘ฅ๐‘–, and since ๐‘ƒ is a regular partition,

๐‘ฅ๐‘–=๐‘ฅ0+๐‘–โกฮ”โข๐‘ฅ=0+๐‘–โก[2๐‘›]=2โข๐‘–๐‘›.

Thus, the function value at the right endpoint of the interval is

๐‘“โก(๐‘ฅ๐‘–)=๐‘ฅ2๐‘–=(2โข๐‘–๐‘›)2=4โข๐‘–2๐‘›2.

Then the Riemann sum takes the form

๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘ฅ๐‘–)โขฮ”โข๐‘ฅ=๐‘›โˆ‘๐‘–=1(4โข๐‘–2๐‘›2)โข2๐‘›=๐‘›โˆ‘๐‘–=18โข๐‘–2๐‘›3=8๐‘›3โข๐‘›โˆ‘๐‘–=1๐‘–2.

Using the summation formula for ๐‘›โˆ‘๐‘–=1๐‘–2, we have

๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘ฅ๐‘–)โขฮ”โข๐‘ฅ=8๐‘›3โข๐‘›โˆ‘๐‘–=1๐‘–2=8๐‘›3โข[๐‘›โข(๐‘›+1)โข(2โข๐‘›+1)6]=8๐‘›3โข[2โข๐‘›3+3โข๐‘›2+๐‘›6]=16โข๐‘›3+24โข๐‘›2+8โข๐‘›6โข๐‘›3=83+4๐‘›+86โข๐‘›2.

Now, to calculate the definite integral, we need to take the limit as ๐‘› โ†’โˆž. We get

โˆซ20๐‘ฅ2๐‘‘๐‘ฅ=lim๐‘›โ†’โˆžโก๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘ฅ๐‘–)โขฮ”โข๐‘ฅ=lim๐‘›โ†’โˆžโก(83+4๐‘›+86โข๐‘›2)=lim๐‘›โ†’โˆžโก(83)+lim๐‘›โ†’โˆžโก(4๐‘›)+lim๐‘›โ†’โˆžโก(86โข๐‘›2)=83+0+0=83.

Exercise 5.2.1

Use the definition of the definite integral to evaluate โˆซ30(2โข๐‘ฅ โˆ’1) ๐‘‘๐‘ฅ.

Use a right-endpoint approximation to generate the Riemann sum.

Hint

Use the solving strategy from Example 5.2.1.

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 ๐‘ฅ-axis.

Example 5.2.2: Using Geometric Formulas to Calculate Definite Integrals

Use the formula for the area of a circle to evaluate โˆซ63โˆš9โˆ’(๐‘ฅโˆ’3)2 ๐‘‘๐‘ฅ.

Solution

The function describes a semicircle with radius 3. To find

โˆซ63โˆš9โˆ’(๐‘ฅโˆ’3)2๐‘‘๐‘ฅ

we want to find the area under the curve over the interval [3,6]. The formula for the area of a circle is ๐ด =๐œ‹โข๐‘Ÿ2. The area of a semicircle is just one-half the area of a circle, or ๐ด =12โข๐œ‹โข๐‘Ÿ2. The shaded area in Figure 5.2.1 covers one-half of the semicircle, or ๐ด =14โข๐œ‹โข๐‘Ÿ2. Thus,

โˆซ63โˆš9โˆ’(๐‘ฅโˆ’3)2๐‘‘๐‘ฅ=14โข๐œ‹โข(3)2=94โข๐œ‹โ‰ˆ7.069.

A graph of a semi circle in quadrant one over the interval [0,6] with center at (3,0). The area under the curve over the interval [3,6] is shaded in blue.
Figure 5.2.1: The value of the integral of the function ๐‘“โก(๐‘ฅ) over the interval [3,6] is the area of the shaded region.
Exercise 5.2.2

Use the formula for the area of a trapezoid to evaluate โˆซ42(2โข๐‘ฅ +3) ๐‘‘๐‘ฅ.

Hint

Graph the function ๐‘“โก(๐‘ฅ) 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 ๐‘“โก(๐‘ฅ) be nonnegative. But how do we interpret โ€œthe area under the curveโ€ when ๐‘“โก(๐‘ฅ) is negative?

Net Signed Area

Let us return to the Riemann sum. Consider, for example, the function ๐‘“โก(๐‘ฅ) =2 โˆ’2โข๐‘ฅ2 (shown in Figure 5.2.2) on the interval [0,2]. Use ๐‘› =8 and choose {๐‘ฅโˆ—๐‘–} as the left endpoint of each interval. Construct a rectangle on each subinterval of height ๐‘“โก(๐‘ฅโˆ—๐‘–) and width ฮ”โข๐‘ฅ. When ๐‘“โก(๐‘ฅโˆ—๐‘–) is positive, the product ๐‘“โก(๐‘ฅโˆ—๐‘–)โขฮ”โข๐‘ฅ represents the area of the rectangle, as before. When ๐‘“โก(๐‘ฅโˆ—๐‘–) is negative, however, the product ๐‘“โก(๐‘ฅโˆ—๐‘–)โขฮ”โข๐‘ฅ represents the negative of the area of the rectangle. The Riemann sum then becomes

8โˆ‘๐‘–=1๐‘“โก(๐‘ฅโˆ—๐‘–)โขฮ”โข๐‘ฅ=(Area of rectangles above the ๐‘ฅ-axis)โˆ’(Area of rectangles below the ๐‘ฅ-axis)

A graph of a downward opening parabola over [-1, 2] with vertex at (0,2) and x-intercepts at (-1,0) and (1,0). Eight rectangles are drawn evenly over [0,2] with heights determined by the value of the function at the left endpoints of each.
Figure 5.2.2: For a function that is partly negative, the Riemann sum is the area of the rectangles above the ๐‘ฅ-axis less the area of the rectangles below the ๐‘ฅ-axis.

Taking the limit as ๐‘› โ†’โˆž, the Riemann sum approaches the area between the curve above the ๐‘ฅ-axis and the ๐‘ฅ-axis, less the area between the curve below the ๐‘ฅ-axis and the ๐‘ฅ-axis, as shown in Figure 5.2.3. Then,

โˆซ20๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ=lim๐‘›โ†’โˆžโก๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘๐‘–)โขฮ”โข๐‘ฅ=๐ด1โˆ’๐ด2.

The quantity ๐ด1 โˆ’๐ด2 is called the net signed area.

A graph of a downward opening parabola over [-2, 2] with vertex at (0,2) and x-intercepts at (-1,0) and (1,0). The area in quadrant one under the curve is shaded blue and labeled A1. The area in quadrant four above the curve and to the left of x=2 is shaded blue and labeled A2.
Figure 5.2.3: In the limit, the definite integral equals area ๐ด1 less area ๐ด2, or the net signed area.

Notice that net signed area can be positive, negative, or zero. If the area above the ๐‘ฅ-axis is larger, the net signed area is positive. If the area below the ๐‘ฅ-axis is larger, the net signed area is negative. If the areas above and below the ๐‘ฅ-axis are equal, the net signed area is zero.

Example 5.2.3: Finding the Net Signed Area

Find the net signed area between the curve of the function ๐‘“โก(๐‘ฅ) =2โข๐‘ฅ and the ๐‘ฅ-axis over the interval [โˆ’3,3].

Solution

The function produces a straight line that forms two triangles: one from ๐‘ฅ =โˆ’3 to ๐‘ฅ =0 and the other from ๐‘ฅ =0 to ๐‘ฅ =3 (Figure 5.2.4). Using the geometric formula for the area of a triangle, ๐ด =12โข๐‘โขโ„Ž, the area of triangle ๐ด1, above the axis, is

๐ด1 =12โข3โข(6) =9,

where 3 is the base and 2โข(3) =6 is the height. The area of triangle ๐ด2, below the axis, is

๐ด2 =12โข(3)โข(6) =9,

where 3 is the base and 6 is the height. Thus, the net area is

โˆซ3โˆ’32โข๐‘ฅ ๐‘‘๐‘ฅ =๐ด1 โˆ’๐ด2 =9 โˆ’9 =0.

A graph of an increasing line over [-6, 6] going through the origin and (-3, -6) and (3,6). The area under the line in quadrant one over [0,3] is shaded blue and labeled A1, and the area above the line in quadrant three over [-3,0] is shaded blue and labeled A2.
Figure 5.2.4: The area above the curve and below the ๐‘ฅ-axis equals the area below the curve and above the ๐‘ฅ-axis.

Analysis

If ๐ด1 is the area above the ๐‘ฅ-axis and ๐ด2 is the area below the ๐‘ฅ-axis, then the net area is ๐ด1 โˆ’๐ด2. Since the areas of the two triangles are equal, the net area is zero.

Exercise 5.2.3

Find the net signed area of ๐‘“โก(๐‘ฅ) =๐‘ฅ โˆ’2 over the interval [0,6], illustrated in the following image.

A graph of an increasing line going through (-2,-4), (0,-2), (2,0), (4,2) and (6,4). The area above the curve in quadrant four is shaded blue and labeled A2, and the area under the curve and to the left of x=6 in quadrant one is shaded and labeled A1.

Hint

Use the solving method described in Example 5.2.3.

Answer

6

Total Area

One application of the definite integral is finding displacement when given a velocity function. If ๐‘ฃโก(๐‘ก) 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, weโ€™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 70 mph for 2 hours, then it is 140 miles away from its original position (Figure 5.2.5). Using integral notation, we have

โˆซ2070๐‘‘๐‘ก=140miles.

A graph in quadrant 1 with the x-axis labeled as t (hours) and y-axis labeled as v (mi/hr). The area under the line v(t) = 75 is shaded blue over [0,2].
Figure 5.2.5: The area under the curve ๐‘ฃโก(๐‘ก) =70 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 miles 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 5.2.6). Again, using integral notation, we have

โˆซ2060๐‘‘๐‘ก+โˆซ52โˆ’40๐‘‘๐‘ก=120โˆ’120=0.

In this case the displacement is zero.

A graph in quadrants one and four with the x-axis labeled as t (hours) and the y axis labeled as v (mi/hr). The first part of the graph is the line v(t) = 60 over [0,2], and the area under the line in quadrant one is shaded. The second part of the graph is the line v(t) = -40 over [2,5], and the area above the line in quadrant four is shaded.
Figure 5.2.6: 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 ๐‘ก-axis, 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

โˆซ20|60|๐‘‘๐‘ก+โˆซ52|โˆ’40|๐‘‘๐‘ก=โˆซ2060๐‘‘๐‘ก+โˆซ5240๐‘‘๐‘ก=120+120=240.

Bringing these ideas together formally, we state the following definitions.

Definition: Net Signed Area

Let ๐‘“โก(๐‘ฅ) be an integrable function defined on an interval [๐‘Ž,๐‘]. Let ๐ด1 represent the area between ๐‘“โก(๐‘ฅ) and the ๐‘ฅ-axis that lies above the axis and let ๐ด2 represent the area between ๐‘“โก(๐‘ฅ) and the ๐‘ฅ-axis that lies below the axis. Then, the net signed area between ๐‘“โก(๐‘ฅ) and the ๐‘ฅ-axis is given by

โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ=๐ด1โˆ’๐ด2.

The total area between ๐‘“โก(๐‘ฅ) and the ๐‘ฅ-axis is given by

โˆซ๐‘๐‘Ž|๐‘“โก(๐‘ฅ)|๐‘‘๐‘ฅ=๐ด1+๐ด2.

Example 5.2.4: Finding the Total Area

Find the total area between ๐‘“โก(๐‘ฅ) =๐‘ฅ โˆ’2 and the ๐‘ฅ-axis over the interval [0,6].

Solution

Calculate the ๐‘ฅ-intercept as (2,0) (set ๐‘ฆ =0, solve for ๐‘ฅ). To find the total area, take the area below the ๐‘ฅ-axis over the subinterval [0,2] and add it to the area above the ๐‘ฅ-axis on the subinterval [2,6] (Figure 5.2.7).

A graph of a increasing line f(x) = x-2 going through the points (-2,-4), (0,2), (2,0), (4,2), and (6,4). The area under the line in quadrant one and to the left of the line x=6 is shaded and labeled A1. The area above the line in quadrant four is shaded and labeled A2.
Figure 5.2.7: The total area between the line and the ๐‘ฅ-axis over [0,6] is ๐ด2 plus ๐ด1.

We have

โˆซ60|(๐‘ฅโˆ’2)|๐‘‘๐‘ฅ=๐ด2+๐ด1.

Then, using the formula for the area of a triangle, we obtain

๐ด2=12โข๐‘โขโ„Ž=12โ‹…2โ‹…2=2

๐ด1=12โข๐‘โขโ„Ž=12โ‹…4โ‹…4=8.

The total area, then, is

๐ด1+๐ด2=8+2=10units2.

Exercise 5.2.4

Find the total area between the function ๐‘“โก(๐‘ฅ) =2โข๐‘ฅ and the ๐‘ฅ-axis over the interval [โˆ’3,3].

Hint

Review the solving strategy in Example 5.2.4.

Answer

18 units2

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. โˆซ๐‘Ž๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ=0(5.2.1)

If the limits of integration are the same, the integral is just a line and contains no area.

2. โˆซ๐‘Ž๐‘๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ=โˆ’โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ

If the limits are reversed, then place a negative sign in front of the integral.

3. โˆซ๐‘๐‘Ž[๐‘“โก(๐‘ฅ)+๐‘”โก(๐‘ฅ)]๐‘‘๐‘ฅ=โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ+โˆซ๐‘๐‘Ž๐‘”โก(๐‘ฅ)๐‘‘๐‘ฅ

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

4. โˆซ๐‘๐‘Ž[๐‘“โก(๐‘ฅ)โˆ’๐‘”โก(๐‘ฅ)]๐‘‘๐‘ฅ=โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅโˆ’โˆซ๐‘๐‘Ž๐‘”โก(๐‘ฅ)๐‘‘๐‘ฅ

The integral of a difference is the difference of the integrals.

5. โˆซ๐‘๐‘Ž๐‘โข๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ=๐‘โขโˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ

for constant ๐‘. The integral of the product of a constant and a function is equal to the constant multiplied by the integral of the function.

6. โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ=โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ+โˆซ๐‘๐‘๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ

Although this formula normally applies when ๐‘ is between ๐‘Ž and ๐‘, the formula holds for all values of ๐‘Ž, ๐‘, and ๐‘, provided ๐‘“โก(๐‘ฅ) is integrable on the largest interval.

Example 5.2.5: Using the Properties of the Definite Integral

Use the properties of the definite integral to express the definite integral of ๐‘“โก(๐‘ฅ) =โˆ’3โข๐‘ฅ3 +2โข๐‘ฅ +2 over the interval [โˆ’2,1] as the sum of three definite integrals.

Solution

Using integral notation, we have โˆซ1โˆ’2(โˆ’3โข๐‘ฅ3 +2โข๐‘ฅ +2) ๐‘‘๐‘ฅ. We apply properties 3. and 5. to get

โˆซ1โˆ’2(โˆ’3โข๐‘ฅ3+2โข๐‘ฅ+2)๐‘‘๐‘ฅ=โˆซ1โˆ’2โˆ’3โข๐‘ฅ3๐‘‘๐‘ฅ+โˆซ1โˆ’22โข๐‘ฅ๐‘‘๐‘ฅ+โˆซ1โˆ’22๐‘‘๐‘ฅ=โˆ’3โขโˆซ1โˆ’2๐‘ฅ3๐‘‘๐‘ฅ+2โขโˆซ1โˆ’2๐‘ฅ๐‘‘๐‘ฅ+โˆซ1โˆ’22๐‘‘๐‘ฅ.

Exercise 5.2.5

Use the properties of the definite integral to express the definite integral of ๐‘“โก(๐‘ฅ) =6โข๐‘ฅ3 โˆ’4โข๐‘ฅ2 +2โข๐‘ฅ โˆ’3 over the interval [1,3] as the sum of four definite integrals.

Hint

Use the solving strategy from Example 5.2.5 and the properties of definite integrals.

Answer

6โขโˆซ31๐‘ฅ3 ๐‘‘๐‘ฅ โˆ’4โขโˆซ31๐‘ฅ2 ๐‘‘๐‘ฅ +2โขโˆซ31๐‘ฅ ๐‘‘๐‘ฅ โˆ’โˆซ313 ๐‘‘๐‘ฅ

Example 5.2.6: Using the Properties of the Definite Integral

If it is known that โˆซ80๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ =10 and โˆซ50๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ =5, find the value of โˆซ85๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ.

Solution

By property 6,

โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ=โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ+โˆซ๐‘๐‘๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ.

Thus,

โˆซ80๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ=โˆซ50๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ+โˆซ85๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ10=5+โˆซ85๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ5=โˆซ85๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ.

Exercise 5.2.6

If it is known that โˆซ51๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ =โˆ’3 and โˆซ52๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ =4, find the value of โˆซ21๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ.

Hint

Use the solving strategy from Example 5.2.6 and the rule on properties of definite integrals.

Answer

โˆ’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 ๐‘“โก(๐‘ฅ) is above another function ๐‘”โก(๐‘ฅ), then the area between ๐‘“โก(๐‘ฅ) and the ๐‘ฅ-axis is greater than the area between ๐‘”โก(๐‘ฅ) and the ๐‘ฅ-axis. This is true depending on the interval over which the comparison is made. The properties of definite integrals are valid whether ๐‘Ž <๐‘, ๐‘Ž =๐‘, or ๐‘Ž >๐‘. The following properties, however, concern only the case ๐‘Ž โ‰ค๐‘, and are used when we want to compare the sizes of integrals.

Comparison Theorem

i. If ๐‘“โก(๐‘ฅ) โ‰ฅ0 for ๐‘Ž โ‰ค๐‘ฅ โ‰ค๐‘, then

โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅโ‰ฅ0.

ii. If ๐‘“โก(๐‘ฅ) โ‰ฅ๐‘”โก(๐‘ฅ) for ๐‘Ž โ‰ค๐‘ฅ โ‰ค๐‘, then

โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅโ‰ฅโˆซ๐‘๐‘Ž๐‘”โก(๐‘ฅ)๐‘‘๐‘ฅ.

iii. If ๐‘š and ๐‘€ are constants such that ๐‘š โ‰ค๐‘“โก(๐‘ฅ) โ‰ค๐‘€ for ๐‘Ž โ‰ค๐‘ฅ โ‰ค๐‘, then

๐‘šโข(๐‘โˆ’๐‘Ž)โ‰คโˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅโ‰ค๐‘€โข(๐‘โˆ’๐‘Ž).

Example 5.2.7: Comparing Two Functions over a Given Interval

Compare ๐‘“โก(๐‘ฅ) =โˆš1+๐‘ฅ2 and ๐‘”โก(๐‘ฅ) =โˆš1+๐‘ฅ 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, ๐‘“โก(๐‘ฅ) appears to be above ๐‘”โก(๐‘ฅ) 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], ๐‘”โก(๐‘ฅ) is above ๐‘“โก(๐‘ฅ). The two functions intersect at ๐‘ฅ =0 and ๐‘ฅ =1 (Figure 5.2.8).

A graph of the function f(x) = sqrt(1 + x^2) in red and g(x) = sqrt(1 + x) in blue over [-2, 3]. The function f(x) appears above g(x) except over the interval [0,1]. A second, zoomed-in graph shows this interval more clearly.
Figure 5.2.8: (a) The function ๐‘“โก(๐‘ฅ) appears above the function ๐‘”โก(๐‘ฅ) 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], ๐‘”โก(๐‘ฅ) โ‰ฅ๐‘“โก(๐‘ฅ). Comparing the integrals over the specified interval [0,1], we also see that โˆซ10๐‘”โก(๐‘ฅ) ๐‘‘๐‘ฅ โ‰ฅโˆซ10๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ (Figure 5.2.9). The thin, red-shaded area shows just how much difference there is between these two integrals over the interval [0,1].

A graph showing the functions f(x) = sqrt(1 + x^2) and g(x) = sqrt(1 + x) over [-3, 3]. The area under g(x) in quadrant one over [0,1] is shaded. The area under g(x) and f(x) is included in this shaded area. The second, zoomed-in graph shows more clearly that equality between the functions only holds at the endpoints.
Figure 5.2.9: (a) The graph shows that over the interval [0,1],๐‘”โก(๐‘ฅ) โ‰ฅ๐‘“โก(๐‘ฅ), where equality holds only at the endpoints of the interval. (b) Viewing the same graph with a greater zoom shows this more clearly.

Average Value of a Function

We often need to find the average of a set of numbers, such as an average test grade. Suppose you received the following test scores in your algebra class: 89, 90, 56, 78, 100, and 69. Your semester grade is your average of test scores and you want to know what grade to expect. We can find the average by adding all the scores and dividing by the number of scores. In this case, there are six test scores. Thus,

89+90+56+78+100+696=4826โ‰ˆ80.33.

Therefore, your average test grade is approximately 80.33, which translates to a Bโˆ’ at most schools.

Suppose, however, that we have a function ๐‘ฃโก(๐‘ก) that gives us the speed of an object at any time ๐‘ก, and we want to find the objectโ€™s average speed. The function ๐‘ฃโก(๐‘ก) takes on an infinite number of values, so we canโ€™t use the process just described. Fortunately, we can use a definite integral to find the average value of a function such as this.

Let ๐‘“โก(๐‘ฅ) be continuous over the interval [๐‘Ž,๐‘] and let [๐‘Ž,๐‘] be divided into n subintervals of width ฮ”โข๐‘ฅ =(๐‘ โˆ’๐‘Ž)/๐‘›. Choose a representative ๐‘ฅโˆ—๐‘– in each subinterval and calculate ๐‘“โก(๐‘ฅโˆ—๐‘–) for ๐‘– =1,2,โ€ฆ,๐‘›. In other words, consider each ๐‘“โก(๐‘ฅโˆ—๐‘–) as a sampling of the function over each subinterval. The average value of the function may then be approximated as

๐‘“๐‘Žโข๐‘ฃโข๐‘’โ‰ˆ๐‘“โก(๐‘ฅโˆ—1)+๐‘“โก(๐‘ฅโˆ—2)+โ‹ฏ+๐‘“โก(๐‘ฅโˆ—๐‘›)๐‘›,

which is basically the same expression used to calculate the average of discrete values.

But we know ฮ”โข๐‘ฅ =๐‘โˆ’๐‘Ž๐‘›, so ๐‘› =๐‘โˆ’๐‘Žฮ”โข๐‘ฅ, and we get

๐‘“๐‘Žโข๐‘ฃโข๐‘’โ‰ˆ๐‘“โก(๐‘ฅโˆ—1)+๐‘“โก(๐‘ฅโˆ—2)+โ‹ฏ+๐‘“โก(๐‘ฅโˆ—๐‘›)๐‘›=๐‘“โก(๐‘ฅโˆ—1)+๐‘“โก(๐‘ฅโˆ—2)+โ‹ฏ+๐‘“โก(๐‘ฅโˆ—๐‘›)(๐‘โˆ’๐‘Žฮ”โข๐‘ฅ).

Following through with the algebra, the numerator is a sum that is represented as โˆ‘๐‘›๐‘–=1๐‘“โก(๐‘ฅ โˆ—๐‘–), and we are dividing by a fraction. To divide by a fraction, invert the denominator and multiply. Thus, an approximate value for the average value of the function is given by

โˆ‘๐‘›๐‘–=1๐‘“โก(๐‘ฅโˆ—๐‘–)(๐‘โˆ’๐‘Žฮ”โข๐‘ฅ)=(ฮ”โข๐‘ฅ๐‘โˆ’๐‘Ž)โข๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘ฅโˆ—๐‘–)=(1๐‘โˆ’๐‘Ž)โข๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘ฅโˆ—๐‘–)โขฮ”โข๐‘ฅ.

This is a Riemann sum. Then, to get the exact average value, take the limit as ๐‘› goes to infinity. Thus, the average value of a function is given by

1๐‘โˆ’๐‘Žโขlim๐‘›โ†’โˆžโก๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘ฅ๐‘–)โขฮ”โข๐‘ฅ=1๐‘โˆ’๐‘Žโขโˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ.

Definition: Average Value of a Function

Let ๐‘“โก(๐‘ฅ) be continuous over the interval [๐‘Ž,๐‘]. Then, the average value of the function ๐‘“โก(๐‘ฅ) (or ๐‘“๐‘Žโข๐‘ฃโข๐‘’) on [๐‘Ž,๐‘] is given by

๐‘“๐‘Žโข๐‘ฃโข๐‘’=1๐‘โˆ’๐‘Žโขโˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ)๐‘‘๐‘ฅ.(5.2.2)

Example 5.2.8: Finding the Average Value of a Linear Function

Find the average value of ๐‘“โก(๐‘ฅ) =๐‘ฅ +1 over the interval [0,5].

Solution

First, graph the function on the stated interval, as shown in Figure 5.2.10.

A graph in quadrant one showing the shaded area under the function f(x) = x + 1 over [0,5].
Figure 5.2.10:The graph shows the area under the function (๐‘ฅ) =๐‘ฅ +1 over [0,5].

The region is a trapezoid lying on its side, so we can use the area formula for a trapezoid ๐ด =12โขโ„Žโก(๐‘Ž +๐‘), where โ„Ž represents height, and ๐‘Ž and ๐‘ represent the two parallel sides. Then,

โˆซ50๐‘ฅ +1 ๐‘‘๐‘ฅ =12โขโ„Žโก(๐‘Ž +๐‘) =12 โ‹…5 โ‹…(1 +6) =352.

Thus the average value of the function is

15โˆ’0โขโˆซ50๐‘ฅ +1 ๐‘‘๐‘ฅ =15 โ‹…352 =72.

Exercise 5.2.7

Find the average value of ๐‘“โก(๐‘ฅ) =6 โˆ’2โข๐‘ฅ over the interval [0,3].

Hint

Use the average value formula (Equation 5.2.2), and use geometry to evaluate the integral.

Answer

3

Key Concepts

  • The definite integral can be used to calculate net signed area, which is the area above the ๐‘ฅ-axis less the area below the ๐‘ฅ-axis. 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.
  • The average value of a function can be calculated using definite integrals.

Key Equations

  • Definite Integral

โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ =lim๐‘›โ†’โˆžโก๐‘›โˆ‘๐‘–=1๐‘“โก(๐‘ฅโˆ—๐‘–)โขฮ”โข๐‘ฅ

  • Properties of the Definite Integral

โˆซ๐‘Ž๐‘Ž๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ =0

โˆซ๐‘Ž๐‘๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ =โˆ’โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ

โˆซ๐‘๐‘Ž[๐‘“โก(๐‘ฅ) +๐‘”โก(๐‘ฅ)] ๐‘‘๐‘ฅ =โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ +โˆซ๐‘๐‘Ž๐‘”โก(๐‘ฅ) ๐‘‘๐‘ฅ

โˆซ๐‘๐‘Ž[๐‘“โก(๐‘ฅ) โˆ’๐‘”โก(๐‘ฅ)] ๐‘‘๐‘ฅ =โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ โˆ’โˆซ๐‘๐‘Ž๐‘”โก(๐‘ฅ) ๐‘‘๐‘ฅ

โˆซ๐‘๐‘Ž๐‘โข๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ =๐‘โขโˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ, for constant ๐‘

โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ =โˆซ๐‘๐‘Ž๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ +โˆซ๐‘๐‘๐‘“โก(๐‘ฅ) ๐‘‘๐‘ฅ

Glossary

average value of a function
(or ๐‘“๐‘Žโข๐‘ฃโข๐‘’) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval
definite integral
a primary operation of calculus; the area between the curve and the ๐‘ฅ-axis 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 ๐‘› 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 ๐‘ฅ-axis such that the area below the ๐‘ฅ-axis is subtracted from the area above the ๐‘ฅ-axis; the result is the same as the definite integral of the function
total area
total area between a function and the ๐‘ฅ-axis is calculated by adding the area above the ๐‘ฅ-axis and the area below the ๐‘ฅ-axis; 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 ๐‘ฅ, then the function in the integrand is followed by ๐‘‘โข๐‘ฅ

This page titled 5.2: The Definite Integral is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?