AP Calculus AB 4.3 Rates of Change in Applied Contexts Other than Motion - MCQs - Exam Style Questions
No-Calc Question
For what value of \(b\) does the integral \(\displaystyle \int_{1}^{b} x^{2}\,dx\) equal \(\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n}\Big(1+\frac{2k}{n}\Big)^{2}\frac{2}{n}\) ?
(A) \(b=2\) only
(B) \(b=3\) only
(C) \(b\) could be any real number.
(D) There is no such value of \(b\).
▶️ Answer/Explanation
The sum is a right-endpoint Riemann sum for \(f(x)=x^{2}\).
Here \(\Delta x=\dfrac{2}{n}\Rightarrow b-a=2\) and \(x_k^{*}=1+\dfrac{2k}{n}\Rightarrow a=1\).
Thus the interval is \([1,3]\) so \(b=3\).
✅ Answer: (B)
Here \(\Delta x=\dfrac{2}{n}\Rightarrow b-a=2\) and \(x_k^{*}=1+\dfrac{2k}{n}\Rightarrow a=1\).
Thus the interval is \([1,3]\) so \(b=3\).
✅ Answer: (B)