Riemann Sum Program for the TI-83+ and TI-84+

RIEMANN, a program for the TI-83+ and TI-84+, approximates the area under a curve (integral) by calculating a Riemann sum, a sum of areas of simple geometric figures intersecting the curve. The program solves Riemann sums using one of four methods and displays a graph when prompted. The program itself is optimized to smaller than 1 kilobyte to reduce its memory impact. This tutorial covers the theory of Riemann sums and how to use this program.

Theory

Integrals are used in the natural and social sciences, statistics, economics, finance, and more, but oftentimes they are difficult to calculate. One method of calculating integrals is to look at the basic definition of an integral, as the area underneath a curve.

We can approximate this area by taking a series of rectangles and packing them underneath the curve. Unlike a smooth curve, it is easy to calculate the area of a rectangle, (base*height), and the more, smaller rectangles we use, the more accurate the approximation will be. There are actually many different ways of placing rectangles to choose from, and using trapezoids is an even more effective approach, but all of these sums converge to the actual integral. Below are 3 different visual representations of Riemann sums:

You can read more about Riemann sums here.

Using the RIEMANN Program

Download RIEMANN.8xp and send the file to your calculator. If you’re not familiar with this process, you can view a tutorial here. To begin, enter the equation you want to use for calculations in the Y1 variable. The Y-Variables screen can be reached by pressing Y=. If your screen doesn’t look like this, press MODE and make sure “Func” is selected.

Press PRGM, select RIEMANN, and press ENTER to run.

Press ENTER to continue.

When a program is not in use, it is generally a good idea to archive it in order to conserve RAM.

PRGM:RIEMANN
ClrHome
PlotsOff
Disp "ENTER F(X) IN Y1
Disp "
Disp "CONTINUE?
Input "YES(1) NO(2) ",G
If G=2:Stop
Disp "
Disp "DRAW PICTURES?
Input "YES(1) NO(2) ",H
ClrHome
Input "LEFT ENDPOINT? ",A
Input "RIGHT ENDPOINT? ",B
If H=1:Then
A->Xmin:B->Xmax
iPart(((B-A)/20->W
If W=0:Goto Z
Lbl U
seq(X,X,A,B,W->L5
Y1(L5->L6
min(L6->Ymin
If Ymin>0:0->Ymin
max(L6->Ymax
If Ymax<0:0->Ymax
W->Xscl
iPart((abs((Ymax-Ymin)/10->Yscl
ClrList L6,L5
End
Lbl 0
ClrHome
Disp "ENTER CHOICE:
Disp "LEFT RECT (1)
Disp "RIGHT RECT (2)
Disp "TRAPEZOIDS (3)
Input "MIDPT RECT (4) ",R
Lbl 1
ClrDraw
Input "N? ",N
(B-A)/N->W
0->S:1->C
Lbl 2
If R=1:Goto 3
If R=2:Goto 4
If R=3:Goto 3
If R=4:Goto 5
Lbl 3
A+W(C-1->X
X->J:X+W->L
Goto 7
Lbl 4
A+CW->X
X-W->J:X->L
Goto 7
Lbl 5
If H≠1:Then
If N>5:Then
1->Z:W/2->H:A->X
Lbl 8
X+H->X:Y1+S->S
A+ZW->X
IS>(Z,N:Goto 8
SW->S:Goto T
End:End
A+CW-W/2->X
X-W/2->J
X+W/2->L
Goto 7
A->G:G+W->G:G->V
Lbl 9
V->X:Y1->Y:V+W->X:4Y+2Y1+S->S
V+2W->V
If VX:Y1->E
B->X:Y1->F
(W/3)(S+E-F->S
Goto T
Lbl 7
Y1->K:K+S->S
If H=1:Goto D
Lbl I
IS>(C,N
Goto 2
If R=3:Then
A->X:Y1->P
B->X:Y1->Q
S+(Q-P)/2->S
End
W*S->S
Lbl T
Disp "SUM=",S
Pause
ClrHome
Lbl E
Menu("ENTER CHOICE","CHANGE N",1,"CHANGE METHOD",0,"QUIT",F)
Lbl F
Stop
Lbl D
If R=3:Then
Y1(L->M
Else:K->M
End
Line(J,0,J,K
Line(J,K,L,M
Line(L,M,L,0
If C=N:Pause
Goto I
Lbl Z
If B-A≤9:Then
.1->W:Else
1->W:End
Goto U