Definite Integrals on TI83/84
Copyright © 2002–2015 by Stan Brown
Copyright © 2002–2015 by Stan Brown
Summary: Your TI83/84 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 TI83/84 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 TI83/84:
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 ]

When you have graphed a function, you can make the TI83/84 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 TI83/84 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.)
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 TI83/84, 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 TI83/84 computes the accumulation.  Example: 3 [2nd] [π] [÷] 2 [ENTER]

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