What a Riemann Sum Is
A definite integral represents accumulated area or accumulated change, but sometimes you are given only a table of values or a function that is difficult to integrate directly. A Riemann sum gives you a way to approximate the integral by breaking an interval into smaller pieces and replacing the curve with rectangles.
Imagine the graph of a function on the interval from to . Divide that interval into several smaller subintervals. On each subinterval, choose a representative point and use the function value at that point as the height of a rectangle.
The total area of those rectangles is an approximation to the area under the curve.
As the rectangles become narrower and narrower, the approximation gets better. In the limit, the Riemann sum becomes the definite integral itself.
The Basic Formula
Suppose we want to approximate the integral of from to using equal subintervals.
The width of each rectangle is:
If is the point chosen from the subinterval, then the Riemann sum has the general form:
The important idea is simple: width × height, added over all the rectangles.
Left, Right, and Midpoint Sums
The three Riemann sums you are most likely to encounter in AP Calculus are the left sum, right sum, and midpoint sum. They differ only in which point of each subinterval is used to determine the rectangle's height.
Left Riemann Sum
For a left Riemann sum, use the left endpoint of each subinterval.
If the function is increasing, the left endpoint gives a rectangle whose height is generally smaller than the function values farther to the right. Therefore, a left sum tends to underestimate the integral for an increasing function.
Right Riemann Sum
For a right Riemann sum, use the right endpoint of each subinterval.
For an increasing function, the right endpoint gives a larger height, so a right sum tends to overestimate the integral.
Midpoint Riemann Sum
For a midpoint sum, use the midpoint of each subinterval.
Midpoint sums are often more accurate than left or right sums because they use a point from the middle of each interval rather than systematically favoring one side.
A Step-by-Step Method
- Identify the interval. Find the lower and upper bounds, and .
- Determine the number of subintervals. This is usually given as .
- Calculate the width. Use .
- Determine the sample points. Use left endpoints, right endpoints, or midpoints depending on the problem.
- Evaluate the function at each sample point.
- Multiply each height by the width and add the resulting areas.
- Interpret the result as an approximation of the definite integral.
Worked Examples
The table gives values of . Use a left Riemann sum with the four subintervals shown to approximate .
Use a right Riemann sum with four equal subintervals to approximate .
Approximate using a midpoint Riemann sum with four equal subintervals.
Suppose is positive and increasing on . Determine whether a left Riemann sum underestimates or overestimates the definite integral.
Riemann Sums and the Definite Integral
Riemann sums are not just a numerical trick. They provide the mathematical foundation for the definite integral.
If we divide the interval from to into equal pieces, then the width of each piece is . As becomes very large, the rectangles become extremely narrow.
In the limit, the approximation approaches the exact accumulated area:
This is the fundamental connection between the geometric idea of adding rectangles and the analytic idea of integration.
How to Tell Which Sum Will Be High or Low
On AP Calculus questions, you are often asked whether a Riemann sum is an overestimate or underestimate. The key is to look at the behavior of the function.
For an increasing function, left sums generally underestimate and right sums generally overestimate.
For a decreasing function, the situation reverses: left sums generally overestimate and right sums generally underestimate.
Midpoint sums do not follow such a simple rule based only on whether the function is increasing or decreasing. Their error depends more directly on the function's concavity.
Common Mistakes
A left sum uses left endpoints, a right sum uses right endpoints, and a midpoint sum uses the middle of each subinterval. Mixing these up changes the entire approximation.
A Riemann sum is not just the sum of the function values. You must multiply the total height by the width of each rectangle. Think height × width.
If the interval is and there are equal subintervals, then . Do not divide by the number of subintervals incorrectly.
A finite Riemann sum is an approximation. It becomes the exact definite integral only in the limiting process as the number of rectangles approaches infinity.
Practice Problems
Use a left Riemann sum with four equal subintervals to approximate .
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Use a right Riemann sum with four equal subintervals to approximate .
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Use a midpoint Riemann sum with four equal subintervals to approximate .
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If is decreasing on an interval, will a right Riemann sum tend to overestimate or underestimate the definite integral?
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