How to find the area between two curves (with a worked example)

How to · Updated 2026-11-10

Find where the curves cross, take the top curve minus the bottom one, and integrate between the crossing points. Worked example with y = x + 2 and y = x², plus checks.

To find the area between two curves, solve f(x) = g(x) for the crossing points a and b, then integrate (top curve − bottom curve) from a to b; if the curves swap places, split at each crossing and add the pieces as positive numbers. The Area between curves view of the riemann sums page finds the crossings, shades the region and gives the area, so you can check your working.

What you need

• The two equations as y = something in x, such as y = x + 2 and y = x².
• The limits: the crossing points, or two x values from the question.
• Antiderivatives of the terms: the integral of xⁿ is xⁿ⁺¹ / (n + 1).
• Which curve is on top. Put an x between the limits into both; the bigger answer is on top.

Step by step

The formula is

Area = ∫ from a to b of [f(x) − g(x)] dx, where f is the upper curve and g the lower one.

Each thin strip has height f(x) − g(x) and width dx; the integral adds them all. If the curves cross inside the interval, f − g changes sign, so integrate each piece separately and take its size: in effect, integrate |f(x) − g(x)|.

With the tool:

1. Open the Riemann sums page and press Area between curves at the top.
2. Type the first curve in the Type f(x) box and the second in Second curve g(x). Use ^ for powers, so x² is x^2.
3. Leave Use the points where the curves cross as a and b ticked. The page looks for crossings between x = −10 and 10 and fills in From a and To b. Untick it to type limits from your question.
4. Read the line under the graph, then the facts list: it gives the Area between the curves (always positive), the plain ∫ (f − g) dx, and each piece with its start, end and which curve is on top.

A worked example

Find the area enclosed by y = x + 2 and y = x².

• Crossings. Set x² = x + 2, so x² − x − 2 = 0, which factors as (x − 2)(x + 1) = 0. The curves meet at x = −1 and x = 2.
• Which is on top? Try x = 0: the line gives 2 and the parabola gives 0, so the line is above.
• Set up. Area = ∫ from −1 to 2 of (x + 2 − x²) dx.
• Antiderivative. F(x) = x²/2 + 2x − x³/3.
• At x = 2: 2 + 4 − 8/3 = 10/3.
• At x = −1: 1/2 − 2 + 1/3 = −7/6.
• Subtract. 10/3 − (−7/6) = 20/6 + 7/6 = 27/6 = 4.5 square units.

Typed into the tool, x + 2 and x^2 give crossings at −1 and 2 and an area of 4.5, in one piece with f on top.

When the curves swap. Try f = x^3 and g = x. They cross three times, at −1, 0 and 1. From −1 to 0 the cubic is on top, and from 0 to 1 the line is. Each piece has area 0.25, so the enclosed area is 0.5. The plain integral of (x³ − x) from −1 to 1 is 0, because the pieces cancel; the tool shows both numbers, which is why you split at the middle crossing.

Check your answer

• Archimedes’ shortcut. For the parabola y = x² and any straight line that cuts it at x = a and x = b, the area between them is (b − a)³ / 6. Here (2 − (−1))³ / 6 = 27/6 = 4.5, the same as the integral.
• Box it in. The region fits in a 3 by 4 rectangle (x from −1 to 2, y from 0 to 4), so the area must be under 12.
• Numbers. In the Riemann sums view, pick My own… and type x + 2 - x^2 from −1 to 2. The strips close in on 4.5 as n grows.
• Sign. A negative answer means you took bottom minus top; flip the order.

To see both curves on large axes before you start, draw them in the graphing calculator. If the antiderivative is the hard part, the integration technique coach suggests a method and walks through it.

Questions people ask

Do I need to know which curve is on top? It keeps each piece positive. Get it the wrong way round and only the sign flips, so taking the absolute value rescues you.

What if the area is between a curve and the x axis? Use y = 0 as the second curve (type 0 as g(x)). Parts below the axis then count as positive, unlike a plain definite integral.

Can I integrate with respect to y instead? Yes, and it can be easier for a sideways parabola such as x = y²: use right curve minus left curve, with limits on the y axis. The tool works in x, so swap the letters to check it.