Computing ∫₀¹ x² dx from Definition

Real Analysis · Axiom Academy

EXAMPLE Computing from the Definition We evaluate this definite integral directly from the limit of right Riemann sums — no antiderivatives, just the definition. Evaluate the definite integral from the definition — as the limit of right Riemann sums over the regular partition of [0,1] — without using the Fundamental Theorem of Calculus. Right Riemann rectangles under 𝑦 = 𝑥² on [0, 1]. As the number of subintervals 𝑛 grows (4 → 8 → 16), the total area of the rectangles shrinks toward the exact value ⅓. Nicely done — you computed a definite integral straight from its definition, with no antiderivative in sight. Here is what carried the argument: The definition itself: , with the regular partition x_i = i/n and . The key algebraic identity: the sum-of-squares formula turned the sum into a closed form. The result: — and the power rule ( x^3/3 from 0 to 1 ) confirms it. The same definition computes the integral of any Riemann-integrable function, though the algebra can get heavier. The Fundamental Theorem of Calculus is the fast route in practice, but working from the definition is what makes the integral mean something.

This is the written version of the interactive lesson above. See the full Real Analysis course.