Area Between Curves Calculator

The area between two curves is the integral of (top curve − bottom curve). For y = x + 2 and y = x², the curves meet at x = −1 and x = 2, so the area is \(\int_{-1}^{2}\left(x + 2 – x^{2}\right)dx = \frac{9}{2}\).

How to use the area between curves calculator

1. Enter the two functions.
2. Leave the limits blank to use the first and last points where the curves meet, or type your own limits.
3. Press Find the area. The calculator splits the interval wherever the curves cross, so every piece counts as positive area.

Formula

\(A = \int_{a}^{b} \left|f(x) – g(x)\right|\,dx\)

On a piece where f is on top, the integrand is f − g; where g is on top, it is g − f. Without the absolute value, regions below would cancel regions above.

Worked example

Area between \(y = x^{3}\) and \(y = x\): they cross at −1, 0 and 1. On [−1, 0] the cubic is on top; on [0, 1] the line is. \(A = \int_{-1}^{0}(x^{3} – x)dx + \int_{0}^{1}(x – x^{3})dx = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}\). Integrating x³ − x over [−1, 1] directly would give 0.

Tips

• Sketch or graph the curves first to see which one is on top.
• Solve f(x) = g(x) to find the limits when they are not given.
• If the curves are easier to write as x = h(y), integrate with respect to y instead (right − left).

Area between curves practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you find the area between two curves?

Find where they intersect, decide which is on top on each interval, integrate (top − bottom) on each interval, and add the results.

What if the curves cross more than twice?

Split the interval at every crossing. The calculator finds the crossings and does this automatically.

Do I need to give the limits?

No. If you leave them blank, the calculator uses the first and last intersection points between x = −50 and x = 50.

Why is my answer different from the plain integral?

A plain integral of f − g counts area below as negative. Area between curves always adds positive pieces.

Can I get an exact answer?

Yes, when an antiderivative exists and the crossing points are simple numbers. Otherwise you get an accurate decimal.

Sources

• Paul’s Online Math Notes: Area Between Curves
• OpenStax Calculus Volume 1: Areas between Curves

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