Approximating a Riemann Sum (n = 4)

Calculus 1 · Axiom Academy

EXAMPLE Calculating a Riemann Sum Approximating a definite integral with a right Riemann sum of rectangles. Use a right Riemann sum with n = 4 subintervals to approximate the area under f(x) = x^2 from x = 0 to x = 2 — that is, to estimate . Each rectangle's height is set by the curve at its right edge. Because f(x) = x^2 is increasing, every rectangle pokes above the curve — so this sum will overestimate the true area. Nicely done — you just approximated a definite integral by hand with a right Riemann sum. The recipe: slice [a,b] into n equal pieces of width , pick a sample point in each, and add up . Right endpoints: here the samples were x_i = 0.5, 1, 1.5, 2 , giving heights 0.25, 1, 2.25, 4 and the sum 0.5(7.5) = 3.75 . It's an overestimate: f(x) = x^2 is increasing, so right-edge heights sit above the curve. The exact value is , and . Push n higher and the rectangles hug the curve ever more tightly — that limit, as , is exactly what the definite integral means.

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