Area Under Curve Calculator
Computes the exact area between a parabola f(x) = ax² + bx + c and the x-axis over an interval [x₁, x₂]. Use it to evaluate definite integrals for quadratic functions in physics, economics, or calculus coursework.
About this calculator
The area under the parabola f(x) = ax² + bx + c from x₁ to x₂ is found by evaluating the definite integral ∫[x₁ to x₂] (ax² + bx + c) dx. Using the power rule, the antiderivative is F(x) = (a/3)x³ + (b/2)x² + c·x. The area is then A = |F(x₂) − F(x₁)|, i.e. A = |(a/3·x₂³ + b/2·x₂² + c·x₂) − (a/3·x₁³ + b/2·x₁² + c·x₁)|. The absolute value ensures a positive area even when the parabola dips below the x-axis. This technique is the foundation of integral calculus and applies to any quadratic, including upward-opening (a > 0), downward-opening (a < 0), and linear (a = 0) functions.
How to use
Find the area under f(x) = 2x² − 3x + 1 from x = 1 to x = 4. Enter a = 2, b = −3, c = 1, x₁ = 1, x₂ = 4. Compute F(x) = (2/3)x³ − (3/2)x² + x. F(4) = (2/3)·64 − (3/2)·16 + 4 = 42.667 − 24 + 4 = 22.667. F(1) = (2/3) − 1.5 + 1 = 0.167. Area = |22.667 − 0.167| = 22.5 square units. The calculator performs all three evaluations automatically and returns the result instantly.
Frequently asked questions
Why does the area under curve calculator use absolute value?
A definite integral can return a negative value when the function lies below the x-axis over part or all of the integration interval. In that case, the signed area is negative, but the geometric area (the actual enclosed region) is always positive. Taking the absolute value of F(x₂) − F(x₁) gives the total geometric area between the curve and the x-axis. If your function crosses the x-axis within [x₁, x₂], note that this calculator treats the whole region as one piece — for signed or split-interval areas, you would need to integrate each sub-interval separately.
What types of functions can this area under curve calculator handle?
This calculator is designed for quadratic (parabolic) functions of the form f(x) = ax² + bx + c. Setting a = 0 reduces it to a linear function, and setting both a = 0 and b = 0 makes it a constant function — both are handled correctly. It does not support trigonometric, exponential, or rational functions. For those, a general numerical integration tool would be more appropriate.
How does integration calculate the area under a curve mathematically?
Integration sums infinitely many infinitesimally thin rectangular strips under the curve, each of width dx and height f(x). As the strip width approaches zero, this sum converges to the exact area. For polynomials, the power rule provides an exact antiderivative: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. The Fundamental Theorem of Calculus then states that the definite integral equals F(b) − F(a), where F is the antiderivative evaluated at the two limits. This eliminates the need for any approximation when the function is a polynomial.