MAC 2311/2312 Honors Calculus: Riemann sums on TI-83

Riemann Sums

For example, the Left Riemann Sum with n=10 subintervals for f(t) on [a,b] is

S = h [f(a) + f(a+h) + ... + f(a+9h)].

S = h [f(a+h) + f(a+2h) + ... + f(a+10h)],

S = h [f(a+0.5h) + f(a+1.5h) + ... + f(a+9.5h)],.

TI-83 Basics

Here are two ways to easily compute these Riemann integral sums on your TI-83 graphing calculator: as a sum of a list and as a programmed loop. Keep in mind that:

• you define the function Y₁ (enter its formula) in the Equations Editor Y= (upper left button);
• you refer to the list L₂ by typing 2nd 2;
• you refer to the function Y₁ by typing VARS Y-VARS 1;
• you refer to variables by typing the green ALPHA key (example: to refer to the variable N, type ALPHA LOG);
• you assign values to variables using the STO button (lower left), (example: to store 4 in B, type 4 STO ALPHA MATRX -- this is noted below as 4 sto B).
• The function seq() is in the List Ops menu.
• The function sum() is in the List Math menu.
• The commands For() and End are in the Prgm Ctl menu.

Sum of a List

1. Compute the list of the values of x² for the left-ends of all subintervals by typing (recall that Seq is in the List Ops menu):

   seq(X2, X, 2, 3.8, 0.2) sto L2
   

2. Compute the sum of these values and multiply it by the common factor h=(b-a)/n=(4-2)/10=0.2 (recall that Sum is in the List Math menu):

   0.2 * sum (L2)
   

   You should get 17.48 as a result.
3. Now compute the right sum by changing the input in part 1 (use the Entry button) to

   seq(X2, X, 2.2, 4, 0.2) sto L2
   

   and repeating step 2 (use Entry again); you should get 19.88.

Program a Loop

1. Press Prgm New 1: Create New.
2. Name your program RIEM. (If you already have a program with that name, use a different name.)
3. Enter the following lines in the program, pressing ENTER after each line. (Press PRGM 4 to get "For"; press PRGM 7 to get "End".)

   : 0 sto S
   : (B - A) / N sto H
   : For(K,1,N)
   : A + (K-1+V) * H sto X
   : S + Y1 sto S
   : End
   : H * S
   

4. Test your program to compute the Left Riemann Sum for the function f(x)=x² on the interval [2, 4] with 10 subintervals:
   • press QUIT to exit Prgm mode;
   • define Y₁=x² in the Equation Editor;
   • assign the appropriate values to A (left end), B (right end), N (number of subintervals), and V (type of sum):

         2 sto A
         4 sto B
         10 sto N
         0 sto V

   • execute the program by typing Prgm Exec RIEM Enter. You should get 17.48 as a result. Store it in L (sto L) for later use.
5. Now compute the Right Sum by letting V=1 (type 1 sto V and repeat the step above). You should get 19.88 as a result; store it in R for later use.
6. Which sum is larger: Left or Right? Why?
7. The Trapezoid Sum is the average of the Left and Right Riemann Sums. Compute (L + R) / 2; you should get 18.68 as a result; store it in T.
8. Now compute the Midpoint Sum by letting V=0.5; you should get 18.66 as a result. Store it in M.
9. Of course, the exact integral is (4³ - 2³)/3 = 18 + 2/3 = 18.6667. Store it in E. Compute the error of each method by subtracting the exact value E from L, R, T, or M.
10. Compare the errors M-E and T-E. What do you notice? Which method is "more precise" here -- Midpoint or Trapezoid? How much "more precise"?
11. The Simpson rule computes (2 M + T) / 3, which is a weighted average of M and T with weights 2/3 and 1/3. Compute it and compare it to the exact integral. Yes, they are the same -- Simpson's rule gives the exact answer for quadratic functions.
12. Experiment with larger N. How large should N be if you want precision (the difference between the Right and the Left Sum) of 0.001? See 5.1 in the Calculus textbook.
13. Compute all four sums (Left, Right, Trapezoid, and Simpson) and their errors (difference from the exact integral) for N = 20, 40, 100, and 200. Describe the pattern as N gets larger: what happens to the errors in each case? Based on the pattern, make a conjecture about the error in each case for N=1000. Verify it (be patient; it takes a long time for TI-83 to compute sums for N=1000).
14. Test your program on a different function of your choice (do not pick linear or quadratic functions).
    • Pick reasonable values for A and B and compute the exact integral.
    • Pick N and repeat steps 4-13 above for the function of your choice. Is the Simpson rule exact this time?

The "Exact" Integral

Recall that you can have TI-83 compute definite integrals with higher precision (the result that TI-83 returns is still not exact, but more sophisticated methods are used to achieve the high precision). To compute this "exact" value, just define and plot the function (with appropriate window parameters) and select Calc (top row) Int f(x)dx. This value is TI-83's approximation to the limit of the Riemann sums.

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