About the Area Between Curves Calculator
This area between curves calculator finds the region enclosed by two functions y = f(x) and y = g(x). You can give an interval [a, b] yourself, or let the calculator locate where the curves intersect and use those crossing points as the limits — the usual setup in textbook problems such as "find the area enclosed by y = x² and y = x + 2".
Because area must be positive, the tool splits the interval at every crossing point and integrates |f(x) − g(x)| piece by piece, so it never lets a region above cancel a region below. It also reports the signed integral ∫(f − g) dx, the intersection points it found, and a table showing which curve is on top on each piece.
It is meant for calculus students working on applications of integration and for anyone checking an answer. Integrals are computed numerically with adaptive Simpson’s rule to about ten significant digits; intersections are found by scanning the window and refining sign changes, so tangential touches without a crossing are not split (they do not change the area).
With the default inputs, the area between the curves is 4.5. Change any value above to recalculate instantly.
How to use the area between curves calculator
- 1Enter the two functions f(x) and g(x).
- 2Choose "between intersection points" to let the tool find the limits, or a fixed interval.
- 3Set a and b — the search window, or the exact interval.
- 4Read the area, the intersections used and the signed integral.
- 5Check the table to see which curve is on top in each region.
Formula and method
The area between two curves is the integral of the vertical distance between them: the top function minus the bottom function. When one curve stays on top over the whole interval this is just ∫ₐᵇ (f − g) dx. When the curves cross, the top and bottom swap, so the interval is split at the crossing points and the absolute value of each piece is added.
In "intersection" mode the limits are the smallest and largest x where f(x) = g(x) inside the search window, found by scanning for sign changes of f − g and refining by bisection. Each piece is integrated with adaptive Simpson’s rule. The signed integral ∫(f − g) dx is reported too; it equals the area only when f ≥ g throughout.
- f(x), g(x)
- The two curves
- a, b
- Limits of integration (or intersection points)
- A
- Enclosed area (always ≥ 0)
Worked examples
Region enclosed by y = x + 2 and y = x²
The curves meet where x² = x + 2, i.e. x = −1 and x = 2. The line is on top, so A = ∫₋₁² (x + 2 − x²) dx = [x²/2 + 2x − x³/3] from −1 to 2 = 10/3 − (−7/6) = 4.5.
sin x and cos x on [0, π]
The curves cross at x = π/4. Cos is on top on [0, π/4] (area √2 − 1) and sin is on top on [π/4, π] (area √2 + 1). The total area is 2√2 ≈ 2.8284, while the signed integral is only 2.
y = x³ and y = x (three intersections)
x³ = x at x = −1, 0 and 1. Each lobe has area 1/4: x³ is above x on [−1, 0] and below on [0, 1]. The total area is 0.5, but the signed integral cancels to 0 — which is why the absolute value matters.
Square root and x/2 between intersections
√x = x/2 at x = 0 and x = 4. A = ∫₀⁴ (√x − x/2) dx = [⅔x^(3/2) − x²/4] from 0 to 4 = 16/3 − 4 = 4/3 ≈ 1.3333.
Frequently asked questions
How do you find the area between two curves?+
Find where the curves intersect to get the limits, decide which function is on top, then integrate (top − bottom) dx between the limits. If the curves cross inside the interval, split it at each crossing and add the pieces.
Why is my area zero or negative?+
If you integrate f − g without checking which curve is on top, regions where g is higher subtract from the total. A negative result means g is mostly on top; a result near zero can mean equal regions cancelled out. Area always uses |f − g|.
How do I find the intersection points?+
Set f(x) = g(x) and solve. For polynomials this often means factoring, e.g. x² = x + 2 gives (x − 2)(x + 1) = 0. This calculator finds them numerically inside the search window you set.
Can I integrate with respect to y instead?+
Yes — for regions bounded by curves like x = y², it is often easier to write A = ∫ (right − left) dy. To use this tool, swap the roles of x and y: enter the curves as functions of x, treating x as the y-variable.
What if the curves do not intersect?+
Then there is no enclosed region by themselves. You need extra boundaries such as vertical lines x = a and x = b; choose "Fixed interval" and enter those values.