Riemann Sum
In short: Riemann sum estimates the area under a curve by adding the areas of rectangles built on equal slices of the interval, with heights taken from the function at left, right or middle points.
AP Calculus ABCalculusAP Calculus AB Unit 6
What is Riemann Sum?
A Riemann sum trades a curved region for a stack of rectangles whose areas are easy to add. Cut the interval \([a, b]\) into \(n\) slices of width \(\Delta x = \frac{b – a}{n}\), choose a sample point in each slice, and use the function value there as the rectangle height.
The total is \(\sum_{i=1}^{n} f(x_i^*)\,\Delta x\). Left sums use the left end of each slice, right sums use the right end, and midpoint sums use the center. As \(n\) grows, the rectangles hug the curve, and the sums approach the definite integral.
How to spot it
- A table of values is given at equally spaced times, and you must estimate a total such as distance traveled.
- The problem says “use a left, right or midpoint sum with n subintervals”.
- A graph is split into strips and the question asks for an approximate area.
- You are asked whether an estimate is too large or too small for an increasing or decreasing function.
The method
- Compute the width \(\Delta x = \frac{b – a}{n}\) and list the slice endpoints.
- Pick the sample point in each slice (left end, right end or middle) and evaluate \(f\) there.
- Add the heights, then multiply the total by \(\Delta x\).
- Add units and judge over- or underestimate: for an increasing function a left sum is low and a right sum is high.
Riemann Sum: solved example
Problem
A sensor on a model train records its speed in feet per second every 2 seconds: at \(t = 0, 2, 4, 6, 8\) the speeds are 3, 5, 8, 6, 2. Estimate the distance traveled in 8 seconds with a left sum and with a right sum.
Solution
Four slices, each of width \(\Delta t = 2\).
Left sum: \(2(3 + 5 + 8 + 6) = 2(22) = 44\) feet.
Right sum: \(2(5 + 8 + 6 + 2) = 2(21) = 42\) feet. The distance is about 42 to 44 feet, with the two sums bracketing the estimate.
Try it
Question 1
Use a midpoint sum with \(n = 2\) to estimate the area under \(f(x) = x^2\) on \([0, 4]\).
Show the answer
\(\Delta x = 2\), midpoints 1 and 3: \(2(1 + 9) = 20\). The exact area is about 21.33, so the estimate is close.
Question 2
For a decreasing function, is a right sum an overestimate or an underestimate?
Show the answer
An underestimate: each rectangle uses the right end, which is the lowest point on its slice, so the rectangles sit under the curve.
Riemann Sum vs. Definite Integral
A Riemann sum is a finite estimate that depends on the number of slices and the sample points. The definite integral is the exact value that these estimates approach as the slices get infinitely thin. One is an approximation you can compute by hand; the other is a limit.
Keep going
Lessons
Worksheets and quizzes
Related words
- Definite Integral
- Integral
- Fundamental Theorem of Calculus
- Area Between Curves
- Average Value of a Function
Whole dictionary: A–Z · AP Calculus AB · Calculus