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Area Between Two Curves Calculator
Last updated: May 31, 2026
Written by Blake Boege
An area between two curves calculator is a calculus tool that finds the geometric area of the region bounded by two functions f(x) and g(x) over a specified interval [a, b]. It integrates the absolute difference of the two functions, ∫ |f(x) − g(x)| dx, using numerical approximation methods like Composite Simpson's Rule.
Compute the exact or approximated area enclosed between two functions f(x) and g(x) over a given interval with step-by-step calculus.
Quick Answer
Calculate the exact or numerical area enclosed between two mathematical curves f(x) and g(x) over a defined interval [a, b].
Enclosed Area over [0, 1]
0.166667
Definite integral of absolute difference
Function Values Table (Verification)
| x | f(x) | g(x) | |f(x) - g(x)| |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0.25 | 0.25 | 0.0625 | 0.1875 |
| 0.5 | 0.5 | 0.25 | 0.25 |
| 0.75 | 0.75 | 0.5625 | 0.1875 |
| 1 | 1 | 1 | 0 |
Step-by-Step Derivation
Examples
f(x)=x, g(x)=x^2 from 0 to 1
Area = 0.166667 (1/6)
f(x)=sin(x), g(x)=0 from 0 to pi
Area ≈ 2.000000
How it works
The Calculus of Shaded Regions
When computing the area of a region bounded by two functions, we think of the region as being filled with infinitely thin vertical rectangles of width $dx$ and height $|f(x) - g(x)|$. Summing these heights yields the definite integral:
Definite Integral Formula
Area = ∫[a to b] |f(x) − g(x)| dx
To evaluate a standard single-function integral symbolically, check out our standard integral calculator or use our derivative calculator for tangent and rate of change calculations.
Simpson's Rule
This calculator evaluates the integral using **Composite Simpson's Rule** with 1,000 subintervals. Simpson's rule approximates the shape under the curve using parabolic arcs rather than straight trapezoidal lines, which yields extremely high accuracy for smooth curves.
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Frequently asked questions
To find the area between two curves f(x) and g(x) over an interval [a, b], integrate the absolute difference of the functions: ∫ |f(x) − g(x)| dx from a to b. This ensures that the height of the vertical cross-sections is always positive, yielding a positive area.
If f(x) and g(x) cross paths at some point c between a and b, you must split the integral. Determine which function is higher on each sub-interval, integrate them separately, and add the absolute values of the areas: ∫ [a to c] |f(x) − g(x)| dx + ∫ [c to b] |f(x) − g(x)| dx. Our calculator handles this automatically by integrating the absolute difference |f(x) - g(x)| numerically.
No. The geometric area between two curves is defined as a physical region and is always non-negative. By taking the absolute value of the difference |f(x) − g(x)| before integrating, the integrand is always positive or zero.
Finding symbolic antiderivatives for complex, non-polynomial, or absolute value functions can be extremely difficult or impossible. Numerical integration (such as Simpson's Rule) provides highly accurate approximations for a wide variety of functions.
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