Integration
This section is currently under construction.
1. How can I form a Riemann sum on a regular partition using the calculator?
Answer: We give 3 answers to this question.
In answer #1, we will display a program for computing the right point sum. We call the program RSUM (for right point sum).
In answer #2, we will show another way to compute a Riemann sum using the seq instruction.
In answer #3, we will display a program which will simultaneously compute the left point sum, the midpoint sum, the right point sum, the trapezoid rule sum, and the Simpson rule sum.
Answer #1: Enter the following program into the calculator:
:Prompt N
:Prompt A
:Prompt B
:((B-A)/N) ---> D..........(Note: We use the symbol "--->" to
:0 ---> S............................represent the operation "Store".)
:For(I,1,N)
:(A+I*D) ---> X
:(S+Y1*D) ---> S
:End
:Disp S
This program uses the right hand end points as the x* points. Before executing this program, the function f(x) is to be stored in the Y1 function variable. This program has the advantage that the value of N is unrestricted. However, the program may take a long time to run for very large values of N. (Thank you to the Calculus Committee for this program.)
Answer #2. Put the value of a into A, b into B, n into N, and put the function f(x) into the Y1 variable. On the home screen, enter the instruction (B-A)/N -> H , followed by the instruction H*(sum(Y1(seq(A+K*H,K,1 N)))) . (For the midpoint rule, use H*(sum(Y1(seq(A+(K-.5)*H,K,1,N)))) . For left end points, use H*(sum(Y1(seq(A+(K-1)*H,K,1,N)))) . ) These instructions have the advantage that they are compact and do not need to be stored into a program (although they could be stored into a program). However, a disadvantage of these instructions is that N must be less than 1000 in order to guarantee that the generated sequence has no more than 999 numbers, the maximum size for a sequence within the TI83 calculator.
Answer #3. Here is a program which will compute the left point sum, the midpoint sum, the right point sum, the trapezoid rule sum, and the Simpson rule sum. This program will display all these results together so that comparisons on rates of convergence can be made.
In the program, we store the left point sum in the variable L. The right point sum is stored in the variable R. The midpoint sum is stored in M. The trapezoid rule sum is stored in T. The Simpson rule sum is stored in S. Instructions#8,9 accumulate the right point "rectangles", and instructions#10,11 accumulate the midpoint "rectangles". Instruction#13 takes the right point sum, adds a rectangle at x = A, and subtracts a rectangle at x = B. This results in the left point sum. Instruction#14 computes the trapezoid rule sum as an average of left point and right point sums. Instruction#15 computes the Simpson sum as an appropriate weighted average of the midpoint and trapezoid rule sums.
Before executing the program, put f(x) into the Y1 function variable. Then execute the program. At the prompts, enter the values for N, A, and B. The results are displayed on the home screen. We call the program ISUMS (for "integration" sums).
Instruction#1_____Prompt N
Instruction#2_____Prompt A
Instruction#3_____Prompt B
Instruction#4_____( B - A ) / N ---> D............(Note: We use the symbol "--->" to
Instruction#5_____0 ---> R.................................represent the operation "Store".)
Instruction#6_____0 ---> M
Instruction#7_____For ( I , 1 , N )
Instruction#8_____A + I * D ---> X
Instruction#9_____R + Y1 * D ---> R
Instruction#10____X - D / 2 ---> X
Instruction#11____M + Y1 * D ---> M
Instruction#12____End
Instruction#13____R + D * Y1(A) - D * Y1(B) ---> L
Instruction#14____( L + R ) / 2 ---> T
Instruction#15____( 2M + T ) / 3 ---> S
Instruction#16____Disp "L,M,R,T,S"..........(Note: "Quotes" are needed here.)
Instruction#17____Disp L,M,R,T,S
2. How can I evaluate the (numerical) definite integral of f(X) from "a" to "b" using the calculator?
Answer#1: We assume that f(X) is stored in the function variable Y1. Graph the function in such a way that the interval [a,b] is in the graph window in the x direction. Press [2nd] [CALC] [7] to select the numerical integration option. The calculator will return to the graph window and display a blinking cursor on the graph. The calculator will also ask you for the lower limit of integration. Either use the arrow keys to move the cursor to the point on the graph corresponding to x=a, or type in the number "a". This number will appear at the bottom of the screen. Then press [ENTER]. The calculator will then ask for the upper limit of integration. Either move the cursor to the point corresponding to x=b, or type in the number "b". This number will also appear at the bottom of the screen. Press [ENTER], and the calculator will compute the value of the integral. This value will appear at the bottom of the screen, and the region between the graph and the x-axis will be shaded. The value of the integral is also left in the variable "Ans".
Answer#2: On the home screen, enter the instruction fnInt(Y1, X , lower, upper), where "lower" is the value of "a" and "upper" is the value of "b". (The function "fnInt(" can be found in the [MATH] menu.) After entering the instruction, the value of the integral will be appear on the home screen.
3. Is there a program that will show me the rectangles that are being used when computing a Riemann sum?
Answer: Yes. There are a variety of Calculus programs which can be found at a variety of sources. For the TI83 go to www.ti.com .
4. I have a formula for the function f(x). I use theorems to get a formula for F(x), an antiderivative (or integral) of f(x). How can I check my formula for F(x) using a TI83 calculator?
Answer: There are many answers to this question. Here is one possible answer:
Put your formula for F(x) into the Y3 function variable. Put your formula for f(x) into the Y2 function variable. In Y1, enter the expression nDeriv(Y3,X,X). (nDeriv can be found in the [MATH] menu. Y3 is found in the [VARS] menu.) Next, go to [TBLSET] and select "Indpnt:Ask". Finally, press [TABLE] and enter random values for X.
In your table, the columns for Y1 and Y2 should, in general, be the same to about 5 or 6 significant figures. If they are not the same, then your formula for F(x) is probably not correct. (This is because your numerical derivative using F(x) is not giving the same numbers as your function values using f(x).)
5. I have a formula for f(x). However, I do not have a formula for an antiderivative of f(x). How can I use the TI83 calculator to generate a graph of an antiderivative of f(x)?
Answer: Put f(x) into the Y1 function variable. In Y2, enter the expression fnInt(Y1,X,0,X). (fnInt can be found in the [MATH] menu, item #9.) Then graph Y2. In order to speed up the graphing of Y2, you may want to set Xres (in the [WINDOW] menu) to a number higher than 1, such as 3 or 5.
6. How can I approximate a definite integral with a TI83 using the midpoint rule, the trapezoid rule, and Simpson's rule?
Answer: See answer #3 to question #1 above.
Related questions
Q1. How can I clear a drawing from the graph screen?
Q2. How do I enter the variable Y1 onto the home screen?
Q3. How can I use the TI83 calculator to evaluate the Riemann sum for a double integral over a rectangle R, using a regular partition, and using midpoints in the subrectangles?
Q4. How can I use the sequence mode functions u, v or w to generate a sequence, and then store the sequence in a list variable?
Q5. How can I use the "seq" instruction (in the LIST OPS menu) to generate a sequence, and then store the sequence in a list variable?
Q6. How can I "manually" create a list of numbers and store the list in a list variable?
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