AP Calculus AB Riemann Sums and the Definite Integral Worksheet Solutions (PDF)
This riemann sums 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.
This answer key gives a full worked solution for every problem in the riemann sums worksheet. Compare each step, not only the final answer, because AP graders award credit for a correct setup and clear justification. Final answers are in bold. Redo any problem you missed before moving on to the test.
AP Calculus AB · Common Core State Standards: AP CALC AB 6.1, AP CALC AB 6.2, AP CALC AB 6.3
Riemann sums answers: exercises 1 to 10
Exercise 1 – Left Riemann sum
\(\Delta x=1\). Left endpoints \(0,1,2,3\): \(f=1,2,5,10\).
Sum: \(1(1+2+5+10)=18\).
Final answer: \(18\).
Exercise 2 – Right Riemann sum
Right endpoints \(1,2,3,4\): \(f=2,5,10,17\).
Sum: \(1(2+5+10+17)=34\).
Final answer: \(34\).
Exercise 3 – Midpoint Riemann sum
Midpoints \(0.5,1.5,2.5,3.5\): \(f=1.25,3.25,7.25,13.25\).
Sum: \(1(1.25+3.25+7.25+13.25)=25\). (The exact area is \(\frac{76}{3}\approx 25.33\).)
Final answer: \(25\).
Exercise 4 – Trapezoidal sum
Trapezoidal sum \(=\frac{\text{left}+\text{right}}{2}=\frac{18+34}{2}=26\).
Final answer: \(26\).
Exercise 5 – Riemann sums from a table
Each width is \(2\). Left: \(2(3+5+8+12)=56\). Right: \(2(5+8+12+17)=84\). Trapezoidal: \(\frac{56+84}{2}=70\).
Final answer: \(56,\ 84,\ 70\text{ feet}\).
Exercise 6 – Unequal subintervals
Widths: \(1,3,2,4\). Left sum: \(1(2)+3(4)+2(5)+4(9)=2+12+10+36=60\).
Trapezoidal: \(\frac{2+4}{2}(1)+\frac{4+5}{2}(3)+\frac{5+9}{2}(2)+\frac{9+12}{2}(4)=3+13.5+14+42=72.5\).
Final answer: \(\text{left }60;\ \text{trapezoidal }72.5\).
Exercise 7 – Over or under
\(f\) is increasing, so each left rectangle lies below the curve: underestimate.
Left sum: \(1(0+1+\sqrt{2}+\sqrt{3})\approx 0+1+1.414+1.732=4.146\).
Final answer: \(\approx 4.15\text{ (underestimate)}\).
Exercise 8 – Sigma notation
\(\Delta x=\frac{1}{2}\) and the left endpoints are \(x_{i}=\frac{i}{2}\) for \(i=0,1,2,3\).
\(\sum_{i=0}^{3}\left(\frac{i}{2}\right)^{2}\cdot\frac{1}{2}=\frac{1}{2}\left(0+\frac{1}{4}+1+\frac{9}{4}\right)=\frac{1}{2}(3.5)=1.75\).
Final answer: \(1.75\).
Exercise 9 – Limit as an integral
\(\Delta x=\frac{2}{n}\) means the interval has length \(2\). The sample points \(x_{i}=1+\frac{2i}{n}\) start at \(1\) and end at \(3\).
Final answer: \(\int_{1}^{3}x^{2}\,dx\).
Exercise 10 – Triangle area
The region is a triangle with base \(4\) and height \(f(4)=8\).
Area: \(\frac{1}{2}(4)(8)=16\).
Final answer: \(16\).
Exercises 11 and up
Exercise 11 – Semicircle
The graph is the upper half of the circle \(x^{2}+y^{2}=9\) with radius \(3\).
Area: \(\frac{1}{2}\pi(3)^{2}=\frac{9\pi}{2}\).
Final answer: \(\frac{9\pi}{2}\approx 14.14\).
Exercise 12 – Signed area from a graph
From \(-2\) to \(2\): a triangle above the axis with base \(4\) and height \(4\): area \(8\).
From \(2\) to \(4\): a triangle below the axis with base \(2\) and height \(2\): signed area \(-2\).
Final answer: \(8-2=6\).
Exercise 13 – Properties of integrals
\(\int_{0}^{7}f=5+(-2)=3\). \(\int_{7}^{0}f=-3\). \(\int_{0}^{3}2f=2(5)=10\).
Final answer: \(3,\ -3,\ 10\).
Exercise 14 – Net change versus distance
From \(0\) to \(4\) the velocity is positive: triangle with base \(4\), height \(4\): area \(8\). From \(4\) to \(6\) it is negative: triangle base \(2\), height \(2\): signed area \(-2\).
Displacement: \(8-2=6\). Distance: \(8+2=10\).
Final answer: \(\text{displacement }6\text{ m;}\ \text{distance }10\text{ m}\).
Exercise 15 – Rainfall estimate
Width \(0.25\) h. Trapezoids: \(0.25\left[\frac{0+2}{2}+\frac{2+5}{2}+\frac{5+4}{2}+\frac{4+1}{2}\right]=0.25(1+3.5+4.5+2.5)=0.25(11.5)=2.875\).
Final answer: \(\approx 2.88\text{ inches}\).
Exercise 16 – Electricity use
\(2\left[\frac{1.2+2.0}{2}+\frac{2.0+3.4}{2}+\frac{3.4+2.6}{2}\right]=2(1.6+2.7+3.0)=2(7.3)=14.6\).
Final answer: \(14.6\text{ kWh}\).
Exercise 17 – Bracketing the exact value
\(\Delta x=0.5\). Left: \(0.5(0+0.125+1+3.375)=0.5(4.5)=2.25\). Right: \(0.5(0.125+1+3.375+8)=0.5(12.5)=6.25\).
\(2.25 < 4 < 6.25\).
Final answer: \(\text{left }2.25,\ \text{right }6.25\).
Exercise 18 – Limit of right sums
\(\Delta x=\frac{2}{n}\), \(x_{i}=\frac{2i}{n}\). Sum: \(\sum_{i=1}^{n}\frac{2i}{n}\cdot\frac{2}{n}=\frac{4}{n^{2}}\cdot\frac{n(n+1)}{2}=\frac{2(n+1)}{n}=2+\frac{2}{n}\).
As \(n\to\infty\), \(\frac{2}{n}\to 0\).
Final answer: \(2\text{ (the area of the triangle with base 2 and height 2)}\).
Exercise 19 – Decreasing function
\(\Delta x=1\). Left: \(f(1)+f(2)+f(3)+f(4)=8+4+\frac{8}{3}+2=\frac{50}{3}\approx 16.67\). Right: \(f(2)+f(3)+f(4)+f(5)=4+\frac{8}{3}+2+\frac{8}{5}\approx 10.27\).
For a decreasing function, the left sum overestimates and the right sum underestimates. The exact value is \(8\ln 5\approx 12.88\).
Final answer: \(\text{left }\approx 16.67\text{ (over);}\ \text{right }\approx 10.27\text{ (under)}\).
Exercise 20 – Rectangles and trapezoids
- The region is a rectangle of width \(4\) and height \(3\): \(12\).
- The region is a trapezoid with parallel sides \(f(0)=2\) and \(f(4)=6\) and width \(4\): \(\frac{2+6}{2}\cdot 4=16\). \(16\)
Exercise 21 – Highway traffic
Each width is \(1\) hour. \(\frac{120+180}{2}+\frac{180+260}{2}+\frac{260+200}{2}+\frac{200+140}{2}=150+220+230+170=770\).
Final answer: \(770\text{ cars}\).
Key facts behind this riemann sums answers
- 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 answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
Download the riemann sums answers as a PDF with the button below and keep it with your notes.
Download the riemann sums answers as a PDF with the button below and keep it with your notes.
Download the riemann sums answers as a PDF with the button below and keep it with your notes.
Download the riemann sums answers as a PDF with the button below and keep it with your notes.
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