AP Calculus AB Riemann Sums and the Definite Integral Test Solutions and Scoring (PDF)

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

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This riemann sums test answers 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.

Use this answer key to score the riemann sums test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.

AP Calculus AB · Common Core State Standards: AP CALC AB 6.1, AP CALC AB 6.2, AP CALC AB 6.3

Riemann sums test answers: worked solutions

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

\(1(6+5+4+3)=18\).
\(1(5+4+3+2)=14\).
\(\frac{18+14}{2}=16\). The exact area (trapezoid with parallel sides 6 and 2, width 4) is \(\frac{6+2}{2}\cdot 4=16\); the trapezoidal sum is exact for a line.

Left rectangles for f(x) = 6 − x.

Scoring: 1 point per part.

Question 2 – Table estimate [2 pts]

Right sum: \(3(14+15+11+6)=3(46)=138\) liters.

The rate rises and then falls, so we cannot tell from the table alone whether the right sum is high or low; on the interval where \(r\) increases the right sum overestimates and where it decreases it underestimates.

Scoring: 1 point for the sum with units; 1 point for a correct comment about monotonicity.

Question 3 – Over or under [2 pts]

\(f\) is decreasing on \([1,3]\). On each subinterval, the right endpoint gives the smallest value, so each rectangle lies below the curve.

Final answer: underestimate.

Scoring: 1 point for identifying that f is decreasing; 1 point for the conclusion.

Question 4 – Geometry and integrals [3 pts]

The region is a triangle below the axis with base \(4\) and height \(2\): \(-\frac{1}{2}(4)(2)=-4\).
Add the two triangles above the axis on \([-4,-2]\) and \([2,4]\), each \(+2\): \(-4+2+2=0\).

Graph of f(x) = |x| − 2.

Scoring: 1 point for the first integral; 2 points for the second (1 for the triangles, 1 for the total).

Question 5 – Properties [3 pts]

\(7+(-3)=4\).
\(-\int_{4}^{9}f=3\).
\(3(4)-10=2\).

Scoring: 1 point per part.

Question 6 – Accumulated change [2 pts]

Left sum: \(2(8+6+5)=38\) gallons.

Final answer: about 38 gallons (an overestimate, since the rate is decreasing).

Scoring: 1 point for the sum; 1 point for units.

Question 7 – Sigma and limits [3 pts]

\(\sum_{i=1}^{3}i^{2}\cdot 1=1+4+9=14\).
\(\Delta x=\frac{4}{n}\) and \(x_{i}=\frac{4i}{n}\) on \([0,4]\): \(\int_{0}^{4}\sqrt{x}\,dx\).

Scoring: (a) 1 point for the setup, 1 point for the value; (b) 1 point.

Question 8 – Displacement and distance [2 pts]

Positive part from \(0\) to \(3\): triangle \(\frac{1}{2}(3)(3)=4.5\). Negative part from \(3\) to \(5\): \(-\frac{1}{2}(2)(2)=-2\).

Displacement 2.5 m; distance 6.5 m.

Scoring: 1 point for the displacement; 1 point for the distance.

Total score

Total

20 points

Key facts behind this riemann sums test answers

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 answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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