Multiple Integrals

This section is currently under construction.

1. 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?

Answer: Calculus textbooks will often have exercises concerning the evaluation of a double Riemann sum in the chapter on double integrals. Here is a program which will compute the midpoint double Riemann sum. We assume that the rectangle R is defined for X in the interval [A,B] and Y in the interval [C,D]. We also assume that f(X,Y) is stored in the function variable Y₁ as an expression in X and Y. We assume that the X-interval [A,B] is divided into M equal subintervals, and that the Y-interval [C,D] is divided into N equal subintervals. We call the program IIRSUM for double integral Riemann sum.

After storing f(X,Y) into Y₁, execute the program. Instructions #1,2,3,4,5,6 will give prompts for A,B,C,D,M,N. These values should be entered as the prompts appear on the home screen. Instructions #7,8 compute the spacings H and K for the subintervals in the X-interval [A,B] and Y-interval [C,D] respectively. Thus, H and K are the dimensions of the subrectangles. Instructions #9,10 compute the mid-points of the subrectangles. The list L₁ contains the X-coordinates of these midpoints, and the list L₂ contains the Y-coordinates of these midpoints. Instruction #11 initializes the variable S to zero. The variable S is used to accumulate the sum of the function values at the midpoints of the rectangles. Instructions #12,13,14,15 form a loop which computes the function values at the midpoints, sums their values (a row at a time), and accumulates the result into S. Instruction #16 multiplies the function value sum S by HK, the area of a subrectangle, and puts the result back into S. The result is that the Riemann sum is now in the variable S. The value of the Riemann sum is also displayed by instruction #16.

Instruction#1_____Prompt A
Instruction#2_____Prompt B
Instruction#3_____Prompt C
Instruction#4_____Prompt D
Instruction#5_____Prompt M
Instruction#6_____Prompt N
Instruction#7_____( B - A ) / M ---> H............(Note: We use the symbol "--->" to
Instruction#8_____( D - C ) / N ---> K...............represent the operation "Store".)
Instruction#9_____Seq(A + ( I - .5 ) H , I , 1 , M ) ---> L₁
Instruction#10____Seq(C + ( J - .5 ) K , J , 1 , N ) ---> L₂
Instruction#11____0 ---> S
Instruction#12____For ( J , 1 , N )
Instruction#13____L₂(J) ---> Y
Instruction#14____S + sum( Y₁(L₁) ) ---> S
Instruction#15____End
Instruction#16____H K S ---> S

Note: If one wants to form a Riemann sum using the upper right corner points of the subrectangles, then one should replace ( I - .5 ) in instruction #9 and ( J - .5 ) in instruction #10 by ( I - 0. ) and ( J - 0.) respectively. A similar comment holds if one wants to use the upper left corner points, or the lower right corner points, or the lower left corner points.




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