The Definite Integral
The Definite Integral
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.
Summary
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 \([a,b]\text{.}\) In particular, the symbol \(\int_a^b f(x) \, dx\) denotes the definite integral of \(f\) over \([a,b]\text{,}\) and this quantity is defined by the equation
\begin{equation*} \int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x\text{,} \end{equation*}where \(\Delta x = \frac{b-a}{n}\text{,}\) \(x_i = a + i\Delta x\) (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 \([a,b]\text{;}\) in addition, the value of the definite integral is related to what we call the average value of the function on \([a,b]\text{:}\) \(f_{\text{AVG} [a,b]} = \frac{1}{b-a} \cdot \int_a^b f(x) \, dx\text{.}\) In the setting where we consider the integral of a velocity function \(v\text{,}\) \(\int_a^b v(t) \,dt\) measures the exact change in position of the moving object on \([a,b]\text{;}\) when \(v\) is nonnegative, \(\int_a^b v(t) \,dt\) is the object's distance traveled on \([a,b]\text{.}\)
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{equation*} \int_a^b [c f(x) \pm k g(x)] \,dx = c \int_a^b f(x) \,dx \pm k \int_a^b g(x) \,dx \end{equation*}where \(f\) and \(g\) are continuous functions on \([a,b]\) and \(c\) and \(k\) are arbitrary constants.