Riemann Sums - Left Endpoints and Right Endpoints

This paragraph introduces the concept of calculating the area under a curve using rectangles as an approximation. It begins with a discussion of the graph of y equals x squared and focuses on finding the area under the curve from zero to eight. The method involves dividing the curve into subintervals and approximating the area with rectangles, using either the left (lower sum) or right (upper sum) endpoints. The process of calculating the area of each rectangle and summing them up to estimate the total area under the curve is explained in detail. The paragraph also introduces the concept of delta x, which represents the width of each rectangle, and the sigma notation for summing the areas. The example provided uses four rectangles to approximate the area, resulting in an underestimation of the actual area.

This paragraph delves into the specifics of calculating the area under the curve using left endpoint Riemann sums. It starts by creating a number line from zero to eight and dividing it into four subintervals, resulting in five points. The four points chosen for the left endpoints are 0, 2, 4, and 6. The process of calculating the area for each rectangle using the left endpoints is outlined, with the height of each rectangle being the function value at the chosen point. The paragraph then sums up the areas of the individual rectangles to provide an approximation of the total area, which is found to be 112. It is noted that this is an underestimation of the actual area, and the concept of right endpoint Riemann sums is introduced as a complementary approach to obtain an overestimation.

The paragraph discusses the method of averaging the results from left and right endpoint Riemann sums to achieve a more accurate approximation of the area under the curve. It presents the area calculated using the right endpoints, which results in an overestimation of 240. The process of averaging the left and right endpoint areas, resulting in 176, is then explained. The paragraph also introduces the concept of the definite integral as a way to calculate the exact area under the curve, and the integral of x squared from 0 to 8 is evaluated to be approximately 170.67. The paragraph concludes by highlighting the improved accuracy of the approximation when increasing the number of rectangles from four to eight.

This paragraph explores the impact of increasing the number of rectangles used in the approximation process. By doubling the number of rectangles from four to eight, the accuracy of the left and right endpoint Riemann sums is improved. The paragraph provides a detailed calculation of the area using the left endpoints with eight subintervals, resulting in an area of 140. It then calculates the area using the right endpoints, which is found to be 204. The paragraph organizes the information in a table, comparing the areas calculated using left and right endpoints for both n equals 4 and n equals 8, along with their average and the actual answer. It is observed that as n increases, the approximation becomes closer to the actual area, demonstrating the effectiveness of using more rectangles for a more accurate estimation.