### Essential Concepts

- 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]

## Glossary

- 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]