Riemann sums, definite integrals and the fundamental theorem

Learning objectives

Approximate a definite integral with left, right, midpoint and trapezoidal sums over equal or unequal subintervals, working from a formula, a table or a graph

Decide whether an approximation overestimates or underestimates the integral, arguing from whether the integrand is increasing or decreasing for left and right sums, and from concavity for trapezoidal and midpoint sums

Write a Riemann sum in sigma notation, define the definite integral as its limit as the number of subintervals grows without bound, and convert between such a limit and the integral it represents

Apply the properties of definite integrals: reversed limits, a zero-width interval, additivity over adjoining intervals, constant multiples, sums, and the simplifications available for even and odd integrands

Differentiate an accumulation function g(x) = ∫ from a to x of f(t) dt using the first part of the fundamental theorem, applying the chain rule when the upper limit is itself a function of x, and analyze g for extrema, concavity and inflection from the graph of f

Evaluate a definite integral as F(b) − F(a) using the second part of the fundamental theorem, express a function value as an initial value plus the integral of its rate of change, and interpret that integral as the net change in the accumulated quantity, stating its units