The Definite Integral

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.

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 x-values 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 \(n\to \infty .\) 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:

Condensed — the full section is in OpenStax Calculus Volume 2.

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 x-axis.

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?

Let us return to the Riemann sum. Consider, for example, the function \(f(x)=2-2{x}^{2}\) (shown in ) on the interval \([0,2].\) Use \(n=8\) and choose \(\{{x}_{i}^{*}\text{\}}\) 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 \(f({x}_{i}^{*})\text{\Delta }x\) represents the area of the rectangle, as before. When \(f({x}_{i}^{*})\) is negative, however, the product \(f({x}_{i}^{*})\text{\Delta }x\) represents the negative of the area of the rectangle. The Riemann sum then becomes

Taking the limit as \(n\to \infty ,\) the Riemann sum approaches the area between the curve above the x-axis and the x-axis, less the area between the curve below the x-axis and the x-axis, as shown in . Then,

The quantity \({A}_{1}-{A}_{2}\) is called the net signed area.

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

Condensed — the full section is in OpenStax Calculus Volume 2.

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.

Condensed — the full section is in OpenStax Calculus Volume 2.

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,

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

Suppose, however, that we have a function \(v(t)\) that gives us the speed of an object at any time t, and we want to find the object’s average speed. The function \(v(t)\) takes on an infinite number of values, so we can’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 \(f(x)\) be continuous over the interval \([a,b]\) and let \([a,b]\) be divided into n subintervals of width \(\text{\Delta }x=(b-a)\text{/}n.\) Choose a representative \({x}_{i}^{*}\) in each subinterval and calculate \(f({x}_{i}^{*})\) for \(i=1,2\text{,\ldots ,}\ n.\) In other words, consider each \(f({x}_{i}^{*})\) as a sampling of the function over each subinterval. The average value of the function may then be approximated as

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

But we know \(\text{\Delta }x=\frac{b-a}{n},\) so \(n=\frac{b-a}{\text{\Delta }x},\) and we get

Following through with the algebra, the numerator is a sum that is represented as \(\sum _{i=1}^{n}f({x}_{i}^{*}),\) 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

Condensed — the full section is in OpenStax Calculus Volume 2.

• The definite integral can be used to calculate net signed area, which is the area above the x-axis less the area below the x-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.

In the following exercises, express the limits as integrals.

In the following exercises, given L_n or R_n as indicated, express their limits as \(n\to \infty\) as definite integrals, identifying the correct intervals.

In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis.

In the following exercises, evaluate the integral using area formulas.

In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals.

Suppose that \({\int }_{0}^{4}f(x)dx=5\) and \({\int }_{0}^{2}f(x)dx=-3,\) and \({\int }_{0}^{4}g(x)dx=-1\) and \({\int }_{0}^{2}g(x)dx=2.\) In the following exercises, compute the integrals.

In the following exercises, use the identity \({\int }_{\text{-}A}^{A}f(x)dx={\int }_{\text{-}A}^{0}f(x)dx+{\int }_{0}^{A}f(x)dx\) to compute the integrals.

Condensed — the full section is in OpenStax Calculus Volume 2.

In Figure, we see evidence that increasing the number of rectangles in a Riemann sum improves the accuracy of the approximation of the net signed area bounded by the given function.

We therefore explore the natural idea of allowing the number of rectangles to increase without bound. In an effort to compute the exact net signed area we also consider the differences among left, right, and middle Riemann sums and the different results they generate as the value of \(n\) increases. We begin with functions that are exclusively positive on the interval under consideration.

In Preview Activity, we saw that as the number of rectangles got larger and larger, the values of \(L_n\), \(M_n\), and \(R_n\) all grew closer and closer to the same value. It turns out that this occurs for any continuous function on an interval \([a,b]\), and also for a Riemann sum using any point \(x_{i+1}^*\) in the interval \([x_i, x_{i+1}]\). Thus, as we let \(n \to \infty\), it doesn't really matter where we choose to evaluate the function within a given subinterval, because \[\begin{aligned}\end{aligned}\].

That these limits always exist (and share the same value) when \(f\) is continuous It turns out that a function need not be continuous in order to have a definite integral. For our purposes, we assume that the functions we consider are continuous on the interval(s) of interest. It is straightforward to see that any function that is piecewise continuous on an interval of interest will also have a well-defined definite integral. allows us to make the following definition.

We call the symbol \(\int\) the integral sign, the values \(a\) and \(b\) the limits of integration, and the function \(f\) the integrand. The process of determining the real number \(\int_a^b f(x) \, dx\) is called evaluating the definite integral. While there are several different interpretations of the definite integral, for now the most important is that \(\int_a^b f(x) \, dx\) measures the net signed area bounded by \(y = f(x)\) and the \(x\)-axis on the interval \([a,b]\).

For example, if \(f\) is the function pictured in Figure, and \(A_1\), \(A_2\), and \(A_3\) are the exact areas bounded by \(f\) and the \(x\)-axis on the respective intervals \([a,b]\), \([b,c]\), and \([c,d]\), then \[\begin{aligned}\int_a^b f(x) \, dx = A_1, \ \int_b^c f(x) \, dx = -A_2, \\ \int_c^d f(x) \, dx = A_3, \\ \text{ and } \int_a^d f(x) \, dx = A_1 - A_2 + A_3\end{aligned}\].

Condensed — the full section is in Boelkins, Active Calculus.

Regarding the definite integral of a function \(f\) over an interval \([a,b]\) as the net signed area bounded by \(f\) and the \(x\)-axis, we discover several standard properties of the definite integral. It is helpful to remember that the definite integral is defined in terms of Riemann sums, which consist of the areas of rectangles.

For any real number \(a\) and the definite integral \(\int_a^a f(x) \, dx\) it is evident that no area is enclosed, because the interval begins and ends with the same point. Hence,

If \(f\) is a continuous function and \(a\) is a real number, then \(\int_a^a f(x) \,dx = 0\).

Next, we consider the result of subdividing the interval of integration. In Figure, we see that \[\begin{aligned}\int_a^b f(x) \, dx = A_1, \ \int_b^c f(x) \, dx = A_2, \\ \text{and }\int_a^c f(x) \, dx = A_1 + A_2\end{aligned}\], which illustrates the following general rule.

If \(f\) is a continuous function and \(a\), \(b\), and \(c\) are real numbers, then \[\begin{aligned}\end{aligned}\].

While this rule is easy to see if \(a \lt b \lt c\), it in fact holds in general for any values of \(a\), \(b\), and \(c\). Another property of the definite integral states that if we reverse the order of the limits of integration, we change the sign of the integral's value.

If \(f\) is a continuous function and \(a\) and \(b\) are real numbers, then \[\begin{aligned}\end{aligned}\].

This result makes sense because if we integrate from \(a\) to \(b\), then in the defining Riemann sum we set \(\Delta x = \frac{b-a}{n}\), while if we integrate from \(b\) to \(a\), we have \(\Delta x = \frac{a-b}{n} = -\frac{b-a}{n}\), and this is the only change in the sum used to define the integral.

There are two additional useful properties of the definite integral. When we worked with derivative rules in Chapter, we learned the Constant Multiple Rule and the Sum Rule. Recall that the Constant Multiple Rule says that if \(f\) is a differentiable function and \(k\) is a constant, then \[\begin{aligned}\end{aligned}\], and the Sum Rule says that if \(f\) and \(g\) are differentiable functions, then \[\begin{aligned}\end{aligned}\].

These rules are useful because they allow to deal individually with the simplest parts of certain functions by taking advantage of addition and multiplying by a constant. In other words, the process of taking the derivative respects addition and multiplying by constants in the simplest possible way.

Condensed — the full section is in Boelkins, Active Calculus.

One of the most valuable applications of the definite integral is that it provides a way to discuss the average value of a function, even for a function that takes on infinitely many values. Recall that if we wish to take the average of \(n\) numbers \(y_1\), \(y_2\), \(\ldots\), \(y_n\), we compute \[\begin{aligned}\end{aligned}\].

Since integrals arise from Riemann sums in which we add \(n\) values of a function, it should not be surprising that evaluating an integral is similar to averaging the output values of a function. Consider, for instance, the right Riemann sum \(R_n\) of a function \(f\), which is given by \[\begin{aligned}\end{aligned}\].

Since \(\Delta x = \frac{b-a}{n}\), we can thus write \[\begin{aligned}R_n =\mathstrut \amp (f(x_1) + f(x_2) + \cdots + f(x_n))\cdot \frac{b-a}{n} \\ =\mathstrut \amp (b-a) \frac{f(x_1) + f(x_2) + \cdots + f(x_n)}{n}\end{aligned}\].

We see that the right Riemann sum with \(n\) subintervals is just the length of the interval \((b-a)\) times the average of the \(n\) function values found at the right endpoints. And just as with our efforts to compute area, the larger the value of \(n\) we use, the more accurate our average will be. Indeed, we will define the average value of \(f\) on \([a,b]\) to be \[\begin{aligned}\end{aligned}\].

But we also know that for any continuous function \(f\) on \([a,b]\), taking the limit of a Riemann sum leads precisely to the definite integral. That is, \(\lim_{n \to \infty} R_n = \int_a^b f(x) \, dx\), and thus taking the limit as \(n \to \infty\) in Equation, we have that \[\begin{aligned}\end{aligned}\].

Solving Equation for \(f_{\operatorname{AVG} [a,b]}\), we have the following general principle.

If \(f\) is a continuous function on \([a,b]\), then its average value on \([a,b]\) is given by the formula \[\begin{aligned}\end{aligned}\].

Equation tells us another way to interpret the definite integral: the definite integral of a function \(f\) from \(a\) to \(b\) is the length of the interval \((b-a)\) times the average value of the function on the interval. In addition, when the function \(f\) is nonnegative on \([a,b]\), Equation has a natural visual interpretation.

Condensed — the full section is in Boelkins, Active Calculus.

• Any Riemann sum of a continuous function \(f\) on an interval \([a,b]\) provides an estimate of the net signed area bounded by the function and the horizontal axis on the interval. Increasing the number of subintervals in the Riemann sum improves the accuracy of this estimate, and letting the number of subintervals increase without bound results in the values of the corresponding Riemann sums approaching the exact value of the enclosed net signed area.

• When we take the limit of Riemann sums, we arrive at what we call the definite integral of \(f\) over the interval \([a,b]\). In particular, the symbol \(\int_a^b f(x) \, dx\) denotes the definite integral of \(f\) over \([a,b]\), and this quantity is defined by the equation \[\begin{aligned}\end{aligned}\], where \(\Delta x = \frac{b-a}{n}\), \(x_i = a + i\Delta x\) (for \(i = 0, \ldots, n\)), and \(x_i^*\) satisfies \(x_{i-1} \le x_i^* \le x_i\) (for \(i = 1, \ldots, n\)).

• The definite integral \(\int_a^b f(x) \,dx\) measures the exact net signed area bounded by \(f\) and the horizontal axis on \([a,b]\); in addition, the value of the definite integral is related to what we call the average value of the function on \([a,b]\): \(f_{\text{AVG} [a,b]} = \frac{1}{b-a} \cdot \int_a^b f(x) \, dx\). In the setting where we consider the integral of a velocity function \(v\), \(\int_a^b v(t) \,dt\) measures the exact change in position of the moving object on \([a,b]\); when \(v\) is nonnegative, \(\int_a^b v(t) \,dt\) is the object's distance traveled on \([a,b]\).

• The definite integral is a sophisticated sum, and thus has some of the same natural properties that finite sums have. Perhaps most important of these is how the definite integral respects sums and constant multiples of functions, which can be summarized by the rule \[\begin{aligned}\end{aligned}\] where \(f\) and \(g\) are continuous functions on \([a,b]\) and \(c\) and \(k\) are arbitrary constants.

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.

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 x-values 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 \(n\to \infty .\) 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:

Condensed — the full section is in OpenStax Calculus Volume 1.

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 x-axis.

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?

Let us return to the Riemann sum. Consider, for example, the function \(f(x)=2-2{x}^{2}\) (shown in ) on the interval \([0,2].\) Use \(n=8\) and choose \(\{{x}_{i}^{*}\text{\}}\) 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 \(f({x}_{i}^{*})\text{\Delta }x\) represents the area of the rectangle, as before. When \(f({x}_{i}^{*})\) is negative, however, the product \(f({x}_{i}^{*})\text{\Delta }x\) represents the negative of the area of the rectangle. The Riemann sum then becomes

Taking the limit as \(n\to \infty ,\) the Riemann sum approaches the area between the curve above the x-axis and the x-axis, less the area between the curve below the x-axis and the x-axis, as shown in . Then,

The quantity \({A}_{1}-{A}_{2}\) is called the net signed area.

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

Condensed — the full section is in OpenStax Calculus Volume 1.

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.

Condensed — the full section is in OpenStax Calculus Volume 1.

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,

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

Suppose, however, that we have a function \(v(t)\) that gives us the speed of an object at any time t, and we want to find the object’s average speed. The function \(v(t)\) takes on an infinite number of values, so we can’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 \(f(x)\) be continuous over the interval \([a,b]\) and let \([a,b]\) be divided into n subintervals of width \(\text{\Delta }x=(b-a)\text{/}n.\) Choose a representative \({x}_{i}^{*}\) in each subinterval and calculate \(f({x}_{i}^{*})\) for \(i=1,2\text{,\ldots ,}\ n.\) In other words, consider each \(f({x}_{i}^{*})\) as a sampling of the function over each subinterval. The average value of the function may then be approximated as

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

But we know \(\text{\Delta }x=\frac{b-a}{n},\) so \(n=\frac{b-a}{\text{\Delta }x},\) and we get

Following through with the algebra, the numerator is a sum that is represented as \(\sum _{i=1}^{n}f({x}_{i}^{*}),\) 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

Condensed — the full section is in OpenStax Calculus Volume 1.

• The definite integral can be used to calculate net signed area, which is the area above the x-axis less the area below the x-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.

In the following exercises, express the limits as integrals.

In the following exercises, given L_n or R_n as indicated, express their limits as \(n\to \infty\) as definite integrals, identifying the correct intervals.

In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis.

In the following exercises, evaluate the integral using area formulas.

In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals.

Suppose that \({\int }_{0}^{4}f(x)dx=5\) and \({\int }_{0}^{2}f(x)dx=-3,\) and \({\int }_{0}^{4}g(x)dx=-1\) and \({\int }_{0}^{2}g(x)dx=2.\) In the following exercises, compute the integrals.

In the following exercises, use the identity \({\int }_{\text{-}A}^{A}f(x)dx={\int }_{\text{-}A}^{0}f(x)dx+{\int }_{0}^{A}f(x)dx\) to compute the integrals.

Condensed — the full section is in OpenStax Calculus Volume 1.