Integration & Riemann Sums

To find the area under a curve, fill it with thin rectangles and add them up. Use more and more of them, and the total closes in on the exact integral.

Area by adding up strips

Split [a, b] into n strips of width Δx = (b − a)/n. Each rectangle has area f(x)·Δx, where x is taken at the left edge, right edge or middle of the strip. Their sum is a Riemann sum. As n → ∞ the strips get infinitely thin and the sum becomes the definite integral ∫ₐᵇ f(x) dx.

The shortcut: antiderivatives

The Fundamental Theorem of Calculus says you don't need the rectangles: if F′ = f, then ∫ₐᵇ f(x) dx = F(b) − F(a). Area below the x-axis counts as negative (red). Notice the midpoint and trapezoid rules get very close with far fewer strips than the left or right sums.

Function f(x)

Method

From a = 0

To b = 2

Number of strips n 4

Riemann sum—

Exact ∫—

Error—

Antiderivative—