Area Under Curve Calculator

Computing Area Under a Curve

The area under y=f(x) from x=a to x=b is computed using the definite integral: A = ∫ₐᵇ f(x)dx = F(b)−F(a) where F is any antiderivative of f. This is the Fundamental Theorem of Calculus Part 2. If f(x) is negative on part of [a,b], the integral computes signed area (negative regions subtract). To get total geometric area, integrate |f(x)|.

Area between curves: If f(x) ≥ g(x) on [a,b]: A = ∫ₐᵇ [f(x)−g(x)]dx. First find intersection points (where f=g) to determine a and b, then verify which function is on top.

Common examples: ∫₀¹ x²dx = [x³/3]₀¹ = 1/3. ∫₀^π sin(x)dx = [−cos x]₀^π = −(−1)−(−1) = 2. ∫₁ᵉ ln(x)dx = [x·ln x − x]₁ᵉ = (e−e)−(0−1) = 1. Area between y=x and y=x²: ∫₀¹(x−x²)dx = 1/2−1/3 = 1/6.

Why Area = Integral — The Rigorous Connection

The connection between area and the definite integral is not obvious — it is a theorem. The Riemann integral is defined as the limit of Riemann sums: ∫ₐᵇf(x)dx = lim(n→∞) Σᵢ f(xᵢ*) Δx. Each term f(xᵢ*)Δx is the area of one thin rectangle of width Δx and height f(xᵢ*). The limit of the sum of rectangle areas is, by definition, the area under the curve. So the integral equals the area because the integral was designed to capture exactly this quantity — it's the limiting, infinitely-precise version of "add up the rectangles."

The Fundamental Theorem then gives us a practical way to compute this limit without actually taking limits of sums: find an antiderivative F and compute F(b)−F(a). This is why the FTC is so powerful — it replaces an infinite limiting process with a finite computation.

Signed Area vs Geometric Area — When They Differ

The definite integral computes signed area. Where f(x) > 0, the integral accumulates positive area. Where f(x) < 0, it accumulates negative area. For example, ∫₀^(2π) sin(x)dx = 0, because the positive hump from 0 to π (area = 2) exactly cancels the negative hump from π to 2π (area = −2). This signed interpretation is mathematically correct and physically meaningful — integrating velocity gives displacement, which can be negative if you move backwards, not total distance.

To get geometric (unsigned) area, integrate |f(x)|: find all zeros of f in [a,b], split the integral at each zero, and sum the absolute values of each piece. For sin(x) on [0,2π]: |∫₀^π sin x dx| + |∫_π^(2π) sin x dx| = 2 + 2 = 4.

Consumer and Producer Surplus — Economic Applications

Area between curves has direct economic interpretation. Consumer surplus — the total benefit consumers receive above the market price — equals the area between the demand curve P = D(Q) and the market price P*: CS = ∫₀^(Q*) [D(Q)−P*]dQ. Producer surplus — profit above the minimum sellers would accept — equals the area between the market price and the supply curve: PS = ∫₀^(Q*) [P*−S(Q)]dQ. Total economic welfare (the argument that competitive markets maximise) is CS + PS, the total area between demand and supply curves up to the equilibrium quantity. These are all "area between curves" computations with specific economic meaning.