AP Calculus AB Riemann Sums and the Definite Integral Test with Point Values (PDF)

AP Calculus AB math test: Riemann Sums and the Definite Integral – Maths PDF

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This riemann sums test for AP Calculus AB covers riemann sums and the definite integral (Common Core AP CALC AB 6.1, AP CALC AB 6.2, AP CALC AB 6.3), with a free PDF to print.

This test checks Riemann sums, trapezoidal sums, the definite integral as signed area and accumulation of change (AP Calculus AB, 6.1, 6.2, 6.3). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.

Time and points

About this test

Grade: AP Calculus AB
Chapter: Riemann Sums and the Definite Integral
Standards: AP CALC AB 6.1, AP CALC AB 6.2, AP CALC AB 6.3
Time: 45 minutes
Total: 20 points

Riemann sums test: questions

Question 1 – Riemann sums for a line [3 pts]

Left rectangles for f(x) = 6 − x.

Let \(f(x)=6-x\) on \([0,4]\) with \(n=4\). (a) Find the left sum. (1 point) (b) Find the right sum. (1 point) (c) Find the trapezoidal sum and compare with the exact area. (1 point)

Question 2 – Table estimate [2 pts]

Flow rate of water into a tank.

Water flows into a tank at rate \(r(t)\) liters per minute, shown in the table. Use a right sum with 4 subintervals to estimate the total water in 12 minutes. Is it an over- or underestimate? Explain.

Question 3 – Over or under [2 pts]

For \(f(x)=\frac{1}{x}\) on \([1,3]\), explain whether the right sum with 4 equal subintervals overestimates or underestimates the integral.

Question 4 – Geometry and integrals [3 pts]

Graph of f(x) = |x| − 2.

Use geometry to evaluate (a) \(\int_{-2}^{2}(|x|-2)\,dx\) and (b) \(\int_{-4}^{4}(|x|-2)\,dx\).

Question 5 – Properties [3 pts]

Given \(\int_{1}^{4}f(x)\,dx=7\), \(\int_{4}^{9}f(x)\,dx=-3\) and \(\int_{1}^{9}g(x)\,dx=10\), find (a) \(\int_{1}^{9}f(x)\,dx\), (b) \(\int_{9}^{4}f(x)\,dx\), (c) \(\int_{1}^{9}[3f(x)-g(x)]\,dx\). (1 point each)

Question 6 – Accumulated change [2 pts]

Oil leaks from a tank at \(r(t)\) gallons per hour with \(r(0)=8\), \(r(2)=6\), \(r(4)=5\), \(r(6)=2\). Use a left sum with three subintervals to estimate the oil lost during the first 6 hours.

Question 7 – Sigma and limits [3 pts]

(a) Write the right Riemann sum of \(f(x)=x^{2}\) on \([0,3]\) with \(n=3\) in sigma notation and evaluate it. (2 points) (b) Write \(\lim_{n\to\infty}\sum_{i=1}^{n}\sqrt{\frac{4i}{n}}\cdot\frac{4}{n}\) as a definite integral. (1 point)

Question 8 – Displacement and distance [2 pts]

A runner’s velocity is \(v(t)=3-t\) m/s for \(0\le t\le 5\). Use areas to find the displacement and the total distance.

Key facts behind this riemann sums test

Key facts

Riemann sum: add f(sample point) times the width over all subintervals.
For an increasing function a left sum underestimates and a right sum overestimates; the reverse for a decreasing function.
The trapezoidal sum equals the average of the left and right sums when widths are equal.
The definite integral is the limit of Riemann sums and gives signed area.

How to use this riemann sums test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All AP Calculus AB chapters: AP Calculus AB math.

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