We show that, as the number of subintervals increases indefinitely, the Riemann sum approximation of the area under the graph of f (x) = x2 from 0 to 1 approaches the value 1/3 , which is the exact value of the area.
Partition the interval [0, 1] into n equal subintervals of length Δx = 1>n each, and let x1, x2, … ,xn denote the right endpoints of the subintervals. Let
denote the Riemann sum that estimates the area under the graph of f (x) = x2 on the interval 0 ≤ x ≤ 1.
(a) Show that Sn = 1/ n3 (12 + 22 + … + n2).
(b) Using the previous exercise, conclude that
Sn = n(n + 1)(2n + 1) / 6n3 .
(c) As n increases indefinitely, Sn approaches the area under the curve. Show that this area is 1/3.
S₁ = [f(x₁) + f(x₂) + + f(xn)] Ax
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