BrownMath.com → TI-83/84/89 → Definite Integrals
Updated 22 Dec 2009

# Definite Integrals on TI-83/84

Summary: Your TI-83/84 can compute any definite integral by using a numerical process. That can be a big help to you in checking your work. This page shows you two ways to compute a definite integral with numeric limits, and how to plot an accumulation function. The usual cautions about numerical methods apply, particularly when the function is not well behaved.

## Definite Integrals

### Definite Integrals on the Home Screen

The TI-83/84 computes a definite integral using the `fnint( )` function. To access the function, press the [`MATH`] button and then scroll up or down to find `9:fnint(`.

Example: Suppose you must find the definite integral . By symmetry, that’s , which evaluates to −2(cos π/4 − cos 0) = −2(√2/2 − 1) = 2−√2, approximately 0.5858.

Here’s how to check this on the TI-83/84:

 On the home screen, select `fnint`. [`MATH`] [`9`] First argument: the integrand |sin x| [`MATH`] [`►`] [`1`] for `abs(` [`sin`] [`[x,T,θ,n]`] [`)`] for `sin(x)` [`)`] for the closing parenthesis for `abs(` Second argument: the variable of integration x [`,`] [`x,T,θ,n`] Third argument: the lower limit −π/4 [`,`] [`(-)`] [`2nd` `^` makes `π`] [`÷`] `4` Fourth argument: the upper limit π/4 [`,`] [`2nd` `^` makes `π`] [`÷`] `4` The optional fifth argument, tolerance, is generally not needed. [`)`] [`ENTER`]

### Definite Integrals on the Graph Screen

When you have graphed a function, you can make the TI-83/84 integrate that function numerically on any visible interval. For example, suppose you have graphed |sin x|. To find the integral from −π/4 to π/4, follow these steps:

 Request numerical integration. [`2nd` `F4` makes `CALC`] `7` Answer the “Lower Limit?” prompt. [`(-)`] [`2nd` `^` makes `π`] [`÷`] `4` [`ENTER`] The TI-83/84 marks your lower limit and prompts for an upper limit. Answer the “Upper Limit?” prompt, and read off the approximate value of the integral. [`2nd` `^` makes `π`] [`÷`] `4` [`ENTER`]

(The viewing window for those screen shots is −2π to 2π in the x direction and −2 to 2 in the y direction.)

## Accumulation Functions

An accumulation function is a definite integral where the lower limit of integration is still a constant but the upper limit is a variable. You can graph an accumulation function on your TI-83/84, and find the accumulated value for any x.

For instance, consider . Here’s how to graph it.

 Define the integrand in Y1. (It’s okay to use x as the independent variable; remember that the variable of integration is only a dummy.) [`Y=`] [`MATH`] [`►`] [`1`] [`sin`] [`x,T,θ,n`] [`)`] [`)`] [`ENTER`] Define the accumulation function in Y2. This is `fnint`(integrand,x,0,x). `[MATH] [9]` pastes `fnint(`.  [`VARS`] [`►`] [`1`] [`1`] pastes `Y1`.  Finish the function: `[,] [x,T,θ,n] [,] 0 [,] [x,T,θ,n]` Optional: Cursor to the left of `Y2` and press `[ENTER]` repeatedly to change the line that will trace the accumulation function. Set `Xmin` to the lower limit of integration, and set `Ymin` and `Ymax` to whatever values make sense in the problem. `[WINDOW]`. Here I have chosen −2 to 5 for the y range. Accumulation functions take lots of computation, and that makes them graph very slowly. You can speed up graphing by changing the `Xres` setting (at the cost of a more “bumpy” graph). Now display the graph. Be prepared to wait for quite a while. Press `[GRAPH]` You can use the `Trace` function to find the value of the accumulation function for any desired x. Press `[TRACE]`. Note the function expression in the upper left corner.  Press [`▲`] to trace the accumulation function. (There may be a wait before it is displayed.) Enter the desired x value, and the TI-83/84 computes the accumulation. Example:  `3 [2nd] [π] [÷] 2 [ENTER]`