Calculus: Riemann Rectangle Errors

Objectives

• Learn that for a continuous nonnegative function f, there is one interpretation of the definite integral f(x)dx from a to b, the area of the region R, bounded above by the graph of y = f(x), below by the x-axis, and by the lines x = –a and x = –b
• Visualize and compute values for three different Riemann sums: left-hand endpoint, right-hand endpoint, and midpoint, and use these values to estimate the area of a region R
• Learn about the nature of these estimates as the number of rectangles increases
• Consider other functions and relate these Riemann sums to function characteristics

Vocabulary

• Riemann sum
• left-hand endpoint, right-hand endpoint, and midpoint sum
• area of a plane region
• underestimate, overestimate
• definite integral
• continuous

About the Lesson

This lesson involves three Riemann sums used to estimate the area of a plane region. As a result, students will:

• Conjecture about each estimate as the number of rectangles increases.
• Conjecture about each estimate in relation to certain characteristics of the function.
• Consider the magnitude of the error in each approximation.
• Conjecture about other geometric figures that might produce better estimates.