Definite Integrals on TI-83/84
Copyright © 2002–2022 by Stan Brown, BrownMath.com
Copyright © 2002–2022 by Stan Brown, BrownMath.com
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.
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 ]
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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 ]
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(The viewing window for those screen shots is −2π to 2π in the x direction and −2 to 2 in the y direction.)
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. |
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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). |
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Now display the graph. Be prepared to wait for quite a while. | Press [GRAPH]
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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]
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