Programs to compute Riemann sums on the TI-83 Plus

Remark: this is a short program to do elementary stuff, written after a
few minutes of reading the manual. It appears to work, but I claim no
"expertise" in programming this or any other calculator. Similar
commands (or more efficient ones) probably exist on other calculators of this type,
but you will have to read- the-friendly- manual yourself.

The program below prompts the user for  A,B,N, and computes the left-endpoint
Riemann sum on the interval [A,B], divided into N subintervals, for the function
stored in y1. (Notation: in the description below, --> denotes the STO-> key.)

In PRGM/NEW:

Name: SUMLEFT
: Prompt A,B,N
(the Prompt command is found in the PRGM/ I/O menu.)
: seq (A+(I-1)(B-A)/N, I, 1, N) --> L1
(seq is found in the LIST/OPS menu; this command line generates the
list of  N left endpoints in [A,B] and stores this list in L1)
: sum (((B-A)/N)Y1(L1))--> L
(sum is found in the LIST/MATH menu; this command line computes
the Riemann sum and stores the result in L- this is just a matter of applying
the function y1 to the sequence L1, multiplying the result by (B-A)/N, and adding up
the resulting sequence.)
: Disp "LEFT SUM IS", L
(Disp is found in PRGM/ I/O)
: fnInt(Y1,X,A,B)--> F
(this computes the `exact' value of the integral- only a more efficient approximation, really-
and stores it in F. fnInt is the 9th entry in the MATH/MATH menu.)
: Disp "INT IS", F
: (F-L)/F--> E
(This computes the relative error of the N-interval approximation, compared
to the `exact' value given by the calculator)
: Disp "REL ERROR IS", E

To run the program, go to PRGM/EXEC and choose SUMLEFT.

Exercise: write similar programs SUMRIGHT and SUMMID to compute the right-endpoint
and midpoint Riemann sums with N intervals. Only the second command line needs to be changed,
to
: seq (A+I (B-A)/N, I, 1, N)--> L2
and
: seq (A+(I-1/2)(B-A)/N, I, 1, N)-->L3
respectively. Of course, I am also storing the lists in L2 and L3 now, so
the sum command should be changed accordingly. And I would store the Riemann sums to
R and M, respectively (instead of L); the program line defining the relative error should be changed
accordingly.

When you test your programs with simple functions, you will quickly notice that
in most cases SUMMID has a much smaller relative error (for the same N). N=100 is
a reasonable value to start testing.

PROGRAMMING THE TRAPEZOIDAL RULE

To program SUMTRAP, the following changes are needed:
1)  :seq (A+I(B-A)/N,I,0,N)--> L4  (this is the list register L4, above the 4 key)
2)  : ((B-A)/N)(sum(Y1(L4))-(Y1(A)+Y1(B))/2)-->T (this is the Y1 in VARS/Y-VARS/Function and the same L4 as above)
(You should convince yourself that is the same as the formula in the book.)
3):Disp "TRAPEZOIDAL SUM IS", T
4) Further down,  change the program line defining the relative error to:
: (F-T)/F-->E