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In the preceding section we defined the area under a curve in terms of Riemann sums:
However, this definition came with restrictions. We required $f\left(x\right)$ 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.
The definite integral generalizes the concept of the area under a curve. We lift the requirements that $f\left(x\right)$ be continuous and nonnegative, and define the definite integral as follows.
If $f\left(x\right)$ is a function defined on an interval $\left[a,b\right],$ the definite integral of f from a to b is given by
provided the limit exists. If this limit exists, the function $f\left(x\right)$ is said to be integrable on $\left[a,b\right],$ 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, $\left[a,b\right].$ 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 $f\left(x\right)$ the integrand , and the dx indicates that $f\left(x\right)$ 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:
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