Area Between Two Curves Calculator
Find the area enclosed between two quadratic curves over an interval by numerically integrating the absolute gap between them.
How it works
Area = ∫ from x₀ to x₁ of |f(x) − g(x)| dx · approximated by the composite trapezoidal rule with N sub-intervals
The area between two curves is the integral of the vertical gap between them. Taking the absolute value handles the case where the curves cross inside the interval — each side simply adds positive area rather than cancelling. This calculator evaluates that integral numerically with 2,000 trapezoids, which is accurate to several decimals for smooth quadratics.
Worked example
For f(x) = x and g(x) = x² on [0, 1], the line sits above the parabola, so the gap is x − x². Its integral is ½ − ⅓ = 1/6 ≈ 0.16667, and the numerical estimate matches to five decimals.
Sources: Area between curves — Wikipedia
Frequently asked questions
Why use the absolute value of the difference?
If the curves cross within the interval, the sign of f − g flips. Integrating the raw difference would let the two pieces cancel, understating the geometric area. Taking |f − g| adds each region as positive area, which is what "area between curves" means.
How accurate is the numerical result?
With 2,000 trapezoids the error for smooth quadratics is tiny — typically below 1e-6 of the interval width. For exact answers on polynomials you can still integrate by hand; this tool is for a fast, reliable numeric check.
Can I use straight lines instead of parabolas?
Yes. Set the x² coefficient (a) to 0 and the curve becomes the line b·x + c, so you can mix a line and a parabola or two lines freely.
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This calculator is for educational purposes. Double-check important results before acting on them.