Summary of the Definite Integral

The definite integral can be used to calculate net signed area, which is the area above the [latex]x[/latex]-axis minus the area below the [latex]x[/latex]-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
[latex]\displaystyle\int_a^b f(x) dx = \underset{n\to \infty}{\lim}\underset{i=1}{\overset{n}{\Sigma}} f(x_i^*) \Delta x[/latex]
Properties of the Definite Integral
[latex]\displaystyle\int_a^a f(x) dx = 0[/latex]
[latex]\displaystyle\int_b^a f(x) dx = −\displaystyle\int_a^b f(x) dx[/latex]
[latex]\displaystyle\int_a^b [f(x)+g(x)] dx = \displaystyle\int_a^b f(x) dx + \displaystyle\int_a^b g(x) dx[/latex]
[latex]\displaystyle\int_a^b [f(x)-g(x)] dx = \displaystyle\int_a^b f(x) dx - \displaystyle\int_a^b g(x) dx[/latex]
[latex]\displaystyle\int_a^b cf(x) dx = c \displaystyle\int_a^b f(x) dx[/latex] for constant [latex]c[/latex]
[latex]\displaystyle\int_a^b f(x) dx = \displaystyle\int_a^c f(x) dx + \displaystyle\int_c^b f(x) dx[/latex]

average value of a function: (or [latex]f_{\text{ave}}[/latex]) 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 [latex]x[/latex]-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 [latex]n[/latex] 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 [latex]x[/latex]-axis such that the area below the [latex]x[/latex]-axis is subtracted from the area above the [latex]x[/latex]-axis; the result is the same as the definite integral of the function

total area: total area between a function and the [latex]x[/latex]-axis is calculated by adding the area above the [latex]x[/latex]-axis and the area below the [latex]x[/latex]-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 [latex]x[/latex], then the function in the integrand is followed by [latex]dx[/latex]