AP Calculus AB 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation - MCQs - Exam Style Questions
No-Calc Question
(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)
No-Calc Question
Which of the following limits is equal to \(\displaystyle \int_{2}^{5}x^{2}\,dx\)?
(A) \(\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n}\!\left(2+\frac{k}{n}\right)^{2}\frac{1}{n}\)
(B) \(\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n}\!\left(2+\frac{k}{n}\right)^{2}\frac{3}{n}\)
(C) \(\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n}\!\left(2+\frac{3k}{n}\right)^{2}\frac{1}{n}\)
(D) \(\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n}\!\left(2+\frac{3k}{n}\right)^{2}\frac{3}{n}\)
▶️ Answer/Explanation
On \([2,5]\): \(\Delta x=\dfrac{5-2}{n}=\dfrac{3}{n}\), right endpoints \(x_k=2+\dfrac{3k}{n}\).
Riemann sum: \(\displaystyle \sum_{k=1}^{n}\big(x_k\big)^{2}\,\Delta x=\sum_{k=1}^{n}\!\left(2+\frac{3k}{n}\right)^{2}\frac{3}{n}\).
✅ Answer: (D)