Type a function and the limits a and b to evaluate the definite integral ∫ₐᵇ f(x) dx numerically.
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From the article The Definite Integral as Area — A Visual Explanation.
Question: Find the area under y = x² from x = 0 to x = 3, first approximately with 3 rectangles, then exactly.
Rectangles, right-hand heights (Δx = 1):
heights f(1) = 1, f(2) = 4, f(3) = 9
sum = (1 + 4 + 9) × 1 = 14 — an overestimate, because the curve is rising.
Rectangles, left-hand heights:
heights f(0) = 0, f(1) = 1, f(2) = 4
sum = (0 + 1 + 4) × 1 = 5 — an underestimate.
So the true area is somewhere between 5 and 14. Averaging the two (the trapezium rule) gives (5 + 14)/2 = 9.5, already close.
Exact value:
∫03 x² dx = [ x³/3 ]03 = 3³/3 − 0³/3
= 27/3 − 0 = 9 square units
9 sits neatly between the underestimate 5 and the overestimate 14, exactly as the rectangle picture predicts.
Worked in full in The Definite Integral as Area — A Visual Explanation.
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