The definite integral ∫_a^b f(x) dx represents the signed area between the curve f(x) and the x-axis from x = a to x = b. But how do we calculate it? The answer starts with Riemann sums.
What Is a Riemann Sum?
A Riemann sum approximates the area under a curve by dividing the interval [a, b] into n equal subintervals and drawing rectangles. The sum of all rectangle areas approximates the integral. As n → ∞, the approximation becomes exact.
Left, Right, Midpoint, and Trapezoid
Left sum: rectangle height = f(left endpoint). Right sum: height = f(right endpoint). Midpoint sum: height = f(midpoint) — most accurate for the same n. Trapezoid rule: uses trapezoids instead of rectangles, averaging left and right heights.
The Fundamental Theorem of Calculus
FTC Part 1: If F(x) = ∫_a^x f(t) dt, then F'(x) = f(x). FTC Part 2: ∫_a^b f(x) dx = F(b) − F(a), where F is any antiderivative of f. This connects differentiation and integration.