Riemann Sums (Left, Right, Midpoint)

AP Calculus · Axiom Academy

Lesson Riemann Sums (Left, Right, Midpoint) Approximating area under a curve using rectangles How do you find the area under a curve? The idea: slice the region into thin rectangles, compute each rectangle's area, and add them up. For f(x) on [a,b] divided into n equal subintervals of width : where x_i^* is a sample point in the i th subinterval. Left Riemann Sum: Use the left endpoint of each subinterval. Tends to underestimate for increasing functions. Right Riemann Sum: Use the right endpoint. Tends to overestimate for increasing functions. Midpoint Rule: Use the midpoint of each subinterval. Generally more accurate than left or right. As , all Riemann sums approach the exact area (the definite integral) Left sum underestimates (overestimates) when f is increasing (decreasing) Midpoint is usually more accurate than left or right is the width of each rectangle

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