Area Between Curves Calculator

The area between two curves is the integral of the distance between them: top curve minus bottom curve. If the curves cross, they swap places, so the integral is split at the crossing points. This calculator finds the crossings, integrates exactly when it can, checks the result numerically and shades the region.

Try:

How to use the area between curves calculator

  1. Type the two functions f(x) and g(x).
  2. Enter the start and end of the interval.
  3. Press Calculate to see the area and the shaded graph.

Formula

A=∫ab∣f(x)−g(x)∣ dxA = \int_a^b |f(x) - g(x)|\,dx

Worked example: x + 2 and x² from −1 to 2

  • Upper curve f(x) =: x + 2
  • Lower curve g(x) =: x^2
  • From x =: -1
  • To x =: 2

✓ Answer checked

A=92≈4.5A = \frac{9}{2} \approx 4.5
−1.5−1−0.50.511.522.51234567f(x)g(x)
The shaded region between the two curves is the area.
Step-by-step working (3 steps)

  1. Set up the integral Area between curves

    The area between two curves is the integral of the distance between them, top minus bottom.

    A=∫−12∣f(x)−g(x)∣ dxA = \int_{-1}^{2} \left|f(x) - g(x)\right|\,dx

  2. Integrate the difference

    Find an antiderivative of f(x) − g(x).

    ∫(x−x2+2)dx=(−13)x3+12x2+2x+C\int \left(x - x^{2} + 2\right)dx = \left(-\frac{1}{3}\right)x^{3} + \frac{1}{2}x^{2} + 2x + C

  3. Evaluate each piece and add

    Take the absolute value of each piece so every part counts as positive area.

    A=92≈4.5A = \frac{9}{2} \approx 4.5
Formulas used
A=∫ab∣f(x)−g(x)∣ dxA = \int_a^b |f(x) - g(x)|\,dx
How this was checked
  • ✓ Numerical integration gives 4.5, matching the exact value.

Frequently asked questions

Does it matter which curve is on top?

Not here: the calculator uses the absolute difference, so the area is always positive.

What happens if the curves cross?

The integral is split at each crossing and the pieces are added as positive areas.

What is the area between y = x + 2 and y = x²?

They meet at x = −1 and x = 2, and the area between them is 9/2.

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