AP Calculus AB 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation - MCQs - Exam Style Questions
No-Calc Question
\(\ \displaystyle \lim_{n\to\infty}\sum_{k=1}^{n}\left[\frac{1}{\,2+\dfrac{k}{2n}\,}\cdot\frac{1}{n}\right]=\)
(A) \(\displaystyle \int_{0}^{1/2}\frac{1}{\,2+\tfrac{x}{2}\,}\,dx\)
(B) \(\displaystyle \int_{0}^{1/2}\frac{1}{\,2+x\,}\,dx\)
(C) \(\displaystyle \int_{0}^{1}\frac{1}{\,2+\tfrac{x}{2}\,}\,dx\)
(D) \(\displaystyle \int_{0}^{1}\frac{1}{\,2+x\,}\,dx\)
▶️ Answer/Explanation
View as a Riemann sum with \(\Delta x=\dfrac{1}{n}\) and \(x_k=\dfrac{k}{n}\).
Then the summand is \(f(x_k)\Delta x\) with \(f(x)=\dfrac{1}{\,2+\tfrac{x}{2}\,}\).
Interval is \(x\in[0,1]\).
So the limit equals \(\displaystyle \int_{0}^{1}\frac{1}{\,2+\tfrac{x}{2}\,}\,dx\).
✅ Answer: (C)