MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
For a function f that is continuous on [a,b], which of the following best explains why the limit of any Riemann sum on [a,b] approaches the same value as the norm of the partition approaches 0?
(a) Because the Fundamental Theorem of Calculus ensures that every continuous function has an antiderivative, yielding the same integral value no matter how the area is summed.
(b) Continuity on a closed interval implies uniform continuity, ensuring that the variation of f on each subinterval can be made arbitrarily small.
(c) Because the Extreme Value Theorem gives a maximum and minimum on every subinterval, and these two values become equal as ∥P∥→0, forcing all sums together.
(d) Because the Mean Value Theorem identifies a point in each subinterval where the rectangle height matches the curve exactly, so the sum of these rectangles must converge to a single value.