Home / AP Calculus AB 6.2 Approximating Areas with Riemann Sums – MCQs

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)\(15\)\(9\)\(5\)\(4\)
(A) \(68\)
(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:

\(x\)0246
\(f(x)\)4\(k\)812

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)

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