Module 17  Riemann Sums and the Definite Integral  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 17.2: Lefthand Riemann Sums and the AREA Program  
In Lesson 17.1 you used righthand rectangles to approximate the area of the region bounded by the graph of f(x) = x^{2}, the vertical line x = 1, and the xaxis. In this lesson you will use lefthand Riemann sums to approximate the same area. The sum of the areas of the rectangles shown above is called a lefthand Riemann sum because the lefthand corner of each rectangle is on the curve. Defining the LeftHand Sum Function Suppose that the area under the curve y = f(x) and above the xaxis between the lines x = a and x = b is approximated using lefthand rectangles. A function similar to the one defined in Lesson 17.1 can be used to find the sum of the areas of the lefthand rectangles. Modifying the function in Y_{2} can create the new function.
The first lefthand xcoordinate is found by letting k = 0 and the last lefthand xcoordinate is found by letting k = n – 1. This replaces the first righthand xcoordinate found by letting k = 1 and the last righthand xcoordinate found by letting k = n.
The function Y_{1} = X^{2} should still be defined from Lesson 17.1. The following exploration uses the same procedure that was used to find the righthand Riemann sum with four rectangles, except this time the lefthand Riemann sum function will be used to approximate the area under the curve.
The lefthand Riemann sum with 4 rectangles is approximately 0.21875 square units. Evaluate the lefthand Riemann sum for ten rectangles.
The sum with 10 lefthand rectangles is 0.285 square units. 17.2.1 Evaluate the lefthand Riemann sum using 50 rectangles and using 100 rectangles. Click here for the answer. The areas found using lefthand rectangles appear to converge to the same limit as the areas found using righthand rectangles: 1/3 square unit. When the number of rectangles approaches infinity, the lefthand Riemann function has the same limit as the righthand Riemann function. That is, the area under the curve can be found by using either right or left Reimann sum limits as the number of rectangles approaches infinity because they produce the same result. This is true in general for continuous functions. In fact, one can choose any xvalue in each subinterval to determine the height of the corresponding rectangle and the limit will still be the same as N goes to . Many times the midpoint of each subinterval is used to compute the height of each rectangle. Using midpoints usually reduces the error in the approximation of the area under the curve. The AREA Program
A program created for the TI83 can be used to illustrate the rectangles that approximate the area under a curve. The program AREA draws the rectangles associated with left, right and Downloading the AREA Program
Transferring the Program to the TI83 Click here to get information about how to obtain the needed cable and to review the procedure to transfer the program from your computer to your calculator.
Using the AREA Program
The following procedure illustrates the approximate area of the region bounded by the graph of The function X^{2} should be stored in Y_{1} in the Y= Editor. Make sure that Y1 is selected.
You should see a prompt for the value of A, the left side of the region.
Now you should see a prompt for the value of B, the right side of the region.
Next you see a prompt for the number of subintervals, i.e., the number of rectangles.
The graph of f(x) = x^{2} with the lefthand rectangles will be displayed when you press . The pause indicator (four vertical dots) should appear in the upper righthand corner of the screen. The screen above illustrates using lefthand rectangles to approximate the area under the curve. Viewing the Subsequent Screens Subsequent screens of the program display the approximations obtained using lefthand, righthand, and midpoint Riemann sums, along with illustrations of the rectangles used in the righthand and midpoint Riemann sums.
The lefthand Riemann sum using 10 rectangles is 0.285 square units.
You should see the right Riemann sum, the illustration of midpoint rectangles, and finally the midpoint Riemann sum. The sums of the areas of the ten rectangles used to approximate the area of the region using left, right, and midpoint rectangles are 0.285, 0.385, and 0.3325 square units, respectively. Run the program again several times to illustrate convergence of the Riemann sums. Try 25 rectangles and 50 rectangles. 17.2.2 What happens when you try 100 rectangles? Click here for the answer. The AREA program can be used to find the Riemann sums that approximate the area under different curves and over different intervals on the xaxis. Simply change the function defined in Y_{1} and enter different values for A, B and N.
17.2.3 Use the AREA program on the region under f(x) = 2x + 1 between x = 0 and x = 3 with 6 rectangles, and find the approximate area using lefthand, righthand, and midpoint rectangles. Use the window [0, 3, 1] x [0, 8, 1]. Click here for the answer. 

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