Home / AP Calculus AB 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation – MCQs

AP Calculus AB 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation - MCQs - Exam Style Questions

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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)

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)

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