Riemann Sums and the Fundamental Theorem of Calculus

GRE Math Subject · Axiom Academy

LESSON Riemann Sums and the FTC From discrete approximation to exact integration -- the bridge that connects derivatives and integrals Riemann Sums: The Definition of the Integral Setup: Let f be defined on [a, b]. Partition the interval into n subintervals of width: Choose a sample point in each subinterval. The Riemann sum is: Types of Riemann sums (based on sample point choice): Left sum: (use left endpoint of each subinterval) Right sum: (use right endpoint) For a monotone increasing function: Left sum True area Right sum. For monotone decreasing : the inequalities reverse. The Definite Integral as a Limit Definition: If f is integrable on [a, b], then: This limit exists and is independent of the choice of sample points when f is continuous (or bounded with finitely many discontinuities). GRE application: The GRE sometimes presents limits of sums and asks you to recognize them as definite integrals. Identify the function f, the interval [a, b], and the sample points to convert the sum to an integral. Recognizing Riemann Sums as Integrals Strategy: This is a Riemann sum for f(x) = x³ on [0, 1] with right endpoints. Strategy: Factor out 1/n. Let . Recognize f(x) = sin(x) on [0, ]. Fundamental Theorem of Calculus -- Part 1 FTC Part 1: If f is continuous on [a, b], define the accumulation function: Then F is differentiable on (a, b) and: In words: The derivative of the integral (with variable upper limit) gives back the original function.

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