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Calculator Answers

Area Between Two Curves Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

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.

See also: Integral Calculator, Derivative Calculator, and Double Integral Calculator.

Quick Answer

Calculate the exact or numerical area enclosed between two mathematical curves f(x) and g(x) over a defined interval [a, b].

Area Results

Enclosed Area over [0, 1]

0.166667

Definite integral of absolute difference

Lower bound (a)0
Upper bound (b)1
Simpson intervals (N)1000
Approx. Area0.166667

Function Values Table (Verification)

xf(x)g(x)|f(x) - g(x)|
0000
0.250.250.06250.1875
0.50.50.250.25
0.750.750.56250.1875
1110

Step-by-Step Derivation

[1]Define the area integral: Area = ∫ [0 to 1] |f(x) - g(x)| dx
[2]Let f(x) = x and g(x) = x^2
[3]Evaluate boundary differences:
[4] At x = 0: |f(0) - g(0)| = |0 - 0| = 0
[5] At x = 1: |f(1) - g(1)| = |1 - 1| = 0
[6]Apply Composite Simpson's Rule with N = 1000 intervals (step size h ≈ 0.001):
[7] Area ≈ (h / 3) × [y₀ + 4y₁ + 2y₂ + 4y₃ + ... + y_N]
[8] Computed definite integral area ≈ 0.166667
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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.

Related Calculators

More tools from Education

Integral CalculatorCompute definite integrals of polynomials with the power rule, or numerically with Simpson's rule for safe expressions.Derivative CalculatorDifferentiate polynomials term by term with the power rule — clear breakdown of each term's derivative.Double Integral CalculatorCalculate the definite double integral of a function of two variables over rectangular regions using numerical integration.

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.