Question

Multiple Choice If \(f\) is a positive, continuous function on an interval \([a, b],\) which of the following rectangular approximation methods has a limit equal to the actual area under the curve from \(a\) to \(b\) as the number of rectangles approaches infinity? $$\begin{array}{c}{\text { I. LRAM }} \\ {\text { II. RRAM }} \\ {\text { III. MRAM }}\end{array}$$ (A) I and II only (B) III only (C) I and III only (D)I, II, and III (E) None of these

Short answer

(D) I, II, and III

Step-by-step solution

Understand the Problem

The task is to identify which of the Riemann Sum methods will have a limit equal to the actual area under the curve as the number of rectangles used for the approximation (also known as partitions) tends towards infinity.

Recall the Properties of Riemann Sums

The Left Riemann Sum (LRAM) approximates the area under the curve by using rectangles that touch the curve at their left corners. The Right Riemann Sum (RRAM) does the same but with the rectangles touching the curve at their right corners. The Midpoint Riemann Sum (MRAM) uses rectangles that touch the curve in their middle point.

Apply the Limit Concept

The limit as the number of partitions approaches infinity for all types of Riemann Sum approximations should give the exact area under the curve. In other words, as we use more and more rectangles to approximate the area, our approximation gets better and closer to the actual area under the curve, regardless of the method of approximation.

Choose the Correct Answer

Since the limit as partitions approach infinity for all three methods (LRAM, RRAM, and MRAM) should give the exact area under the curve, the correct answer is (D) I, II, and III.