Question

Multiple Choice If the interval \([0, \pi]\) is divided into \(r\) subintervals of length \(\pi / n\) and \(c_{k}\) is chosen from the \(k\) th subintervals, which of the following is a Riemann sum? (A) \(\sum_{k=1}^{n} \sin \left(c_{k}\right) \quad\) (B) \(\sum_{k=1}^{\infty} \sin \left(c_{k}\right) \quad\) (C) \(\sum_{k=1}^{n} \sin \left(c_{k}\right)\left(\frac{\pi}{n}\right)\) \((\mathbf{D}) \sum_{k=1}^{n} \sin \left(\frac{\pi}{n}\right)\left(c_{k}\right) \quad(\mathbf{E}) \sum_{k=1}^{n} \sin \left(c_{k}\right)\left(\frac{\pi}{k}\right)\)

Short answer

The correct answer is (C) \(\sum_{k=1}^{n} \sin(c_{k}) (\pi/n)\)

Step-by-step solution

Understanding the Riemann Sum

Initially, one needs to understand the format of a Riemann sum. It involves a sum of values of a function at a certain point within each subinterval of its domain and each of these function values is multiplied by the length of that subinterval. The formula for a Riemann sum is ∑f(c_k)Δx, where c_k is a point in the k-th subinterval and Δx is the width of the subinterval.

Identify the Riemann Sum in the Choices

The function here is the sine function, the length of subinterval Δx is given as \( \pi / n \), and the points within the subintervals are given by c_k. So in the context of the options given, we have the function f(c_k) as sin(c_k) and Δx as \( \pi / n \).\ So, a Riemann sum here would involve a sum of the form ∑sin(c_k) * \( \pi / n \). Now, looking at the options given, the only one that fits this format is (C) ∑sin(c_k)*\( \pi / n \).