Definite Integrals on the 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 paren for abs(
Second argument: the variable of integration x	[,] [x,T,θ,n]
Third argument: the lower limit –π/4	[,] [(-)] [2nd] [π] [÷] 4
Fourth argument: the upper limit π/4	[,] [2nd] [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] [CALC] [7]
Answer the “Lower Limit?” prompt.	[(-)] [2nd] [π] [÷] 4 [ENTER]
The TI-83/84 marks your lower limit and prompts for an upper limit.
Answer the “Upper Limit?” prompt.	[2nd] [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

Section 4.4 of L-H-E Calculus 7e introduces the idea of an accumulation function, 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]