Riemann sums
Also known as: riemann sum approximation
A Riemann sum approximates the area under a curve by dividing the interval into subintervals and adding the areas of rectangles built on them. As the rectangles get narrower, the sum approaches the exact value of the definite integral.
A Riemann sum estimates the area between a function and the x-axis over an interval [a, b]. Split the interval into n subintervals, each of width Δx = (b − a) / n, then build a rectangle on each one whose height is the function's value at a chosen sample point. Adding those rectangle areas gives the approximation. The name of the sum comes from where the sample point sits: a left Riemann sum uses the left endpoint of each subinterval, a right Riemann sum uses the right endpoint, and a midpoint sum uses the center.
Take f(x) = x² on [0, 2] with n = 4, so Δx = 0.5. A left sum uses heights at x = 0, 0.5, 1, and 1.5, giving 0.5(0 + 0.25 + 1 + 2.25) = 1.75. A right sum uses heights at x = 0.5, 1, 1.5, and 2, giving 0.5(0.25 + 1 + 2.25 + 4) = 3.75. The exact value is 8/3 ≈ 2.67, so both are rough, and they bracket the true answer.
That bracketing follows a rule worth memorizing: on an increasing function, a left sum underestimates and a right sum overestimates; on a decreasing function, the relationship reverses. Concavity governs the trapezoidal rule, a close relative that replaces rectangles with trapezoids — it overestimates when the function is concave up and underestimates when it is concave down.
Riemann sums matter because they define the definite integral. As n approaches infinity and Δx approaches zero, the sum converges to the integral of f from a to b, which is why integral notation resembles an elongated S for "sum." AP Calculus AB tests them constantly, especially in problems that supply data in a table rather than a formula: you are asked to approximate an integral with left, right, midpoint, or trapezoidal sums and then state whether the estimate is an overestimate or underestimate.
Key takeaways
- A Riemann sum approximates area under a curve using rectangles of width Δx = (b − a) / n.
- Left, right, and midpoint sums differ only in where the rectangle height is sampled.
- For an increasing function, left sums underestimate and right sums overestimate; the reverse holds when decreasing.
- The trapezoidal rule overestimates for concave-up functions and underestimates for concave-down functions.
- The definite integral is the limit of a Riemann sum as the number of subintervals approaches infinity.