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  • 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:

A = lim n i = 1 n f ( x i * ) Δ x .

However, this definition came with restrictions. We required f ( x ) to be continuous and nonnegative. Unfortunately, real-world 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

If f ( x ) is a function defined on an interval [ a , b ] , the definite integral    of f from a to b is given by

a b f ( x ) d x = lim n i = 1 n f ( x i * ) Δ x ,

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 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 . 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:

a b f ( x ) d x = a b f ( t ) d t = a b f ( u ) d u

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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