AP Calculus AB 6.2 Approximating Areas with Riemann Sums - MCQs - Exam Style Questions
No-Calc Question
On a certain day, the rate at which material is deposited at a recycling center is modeled by the function \(R\), where \(R(t)\) is measured in tons per hour and \(t\) is the number of hours since the center opened. Using a trapezoidal sum with the three subintervals indicated by the data in the table, what is the approximate number of tons of material deposited in the first \(9\) hours since the center opened?
| \(t\) (hours) | \(0\) | \(2\) | \(7\) | \(9\) |
|---|---|---|---|---|
| \(R(t)\) (tons per hour) | \(15\) | \(9\) | \(5\) | \(4\) |
(B) \(70.5\)
(C) \(85\)
(D) \(136\)
▶️ Answer/Explanation
Trapezoidal sum over \([0,2],\,[2,7],\,[7,9]\):
\(\displaystyle \frac{R(0)+R(2)}{2}\times(2-0)\;+\;\frac{R(2)+R(7)}{2}\times(7-2)\;+\;\frac{R(7)+R(9)}{2}\times(9-7)\)
\(=\dfrac{15+9}{2}\times 2\;+\;\dfrac{9+5}{2}\times 5\;+\;\dfrac{5+4}{2}\times 2\)
\(=12\times 2\;+\;7\times 5\;+\;\dfrac{9}{2}\times 2\)
\(=24\;+\;35\;+\;9\;=\;68\)
✅ Answer: (A) \(68\)
No-Calc Question
The function \(f\) is continuous on \([0,6]\) and has the values in the table:
| \(x\) | 0 | 2 | 4 | 6 |
| \(f(x)\) | 4 | \(k\) | 8 | 12 |
Using the composite trapezoidal rule with \(3\) equal subintervals gives \(\displaystyle \int_{0}^{6} f(x)\,dx \approx 52\). What is the value of \(k\)?
(A) \(2\)
(B) \(6\)
(C) \(7\)
(D) \(10\)
(E) \(14\)
▶️ Answer/Explanation
With \(\Delta x=2\), the trapezoidal rule gives \[ \int_{0}^{6} f(x)\,dx \approx \frac{\Delta x}{2}\Bigl(f(0)+2f(2)+2f(4)+f(6)\Bigr) =\frac{2}{2}\bigl(4+2k+2\times 8+12\bigr) =32+2k. \] Set \(32+2k=52\Rightarrow k=10\).
✅ Answer: (D)