Area Between Two Curves Integral Calculator

The area between two curves is a fundamental concept in calculus that measures the space enclosed by two functions over a specific interval. This calculator helps you compute this area accurately using definite integrals.

What is the Area Between Two Curves?

The area between two curves is the region enclosed by two functions, y = f(x) and y = g(x), from x = a to x = b. This area can be calculated using definite integrals, which sum up the area of infinitely thin vertical strips between the curves.

This concept is essential in physics, engineering, and economics for analyzing quantities like work done by a variable force, the area under a velocity-time graph, or the profit between two cost functions.

How to Calculate the Area Between Two Curves

To find the area between two curves:

Identify the upper and lower functions (f(x) and g(x))
Determine the interval [a, b] where the curves intersect or the area is bounded
Set up the integral as ∫[a to b] (f(x) - g(x)) dx
Evaluate the integral to find the area

If the curves cross within the interval, you'll need to split the integral at the point of intersection.

The Formula

The area A between two curves y = f(x) and y = g(x) from x = a to x = b is given by:

A = ∫[a to b] (f(x) - g(x)) dx

If the curves cross within the interval, the integral becomes:

A = ∫[a to c] (f(x) - g(x)) dx + ∫[c to b] (g(x) - f(x)) dx

Where c is the point of intersection between the curves.

Worked Example

Find the area between the curves y = x² and y = 2x from x = 0 to x = 2.

First, identify which function is upper and which is lower. At x = 1, y = x² = 1 and y = 2x = 2, so 2x is above x² in this interval.

A = ∫[0 to 2] (2x - x²) dx

= [x² - (x³)/3] evaluated from 0 to 2

= (4 - 8/3) - (0 - 0)

= 4/3 ≈ 1.333

The area between the curves is 4/3 square units.

FAQ

What if the curves cross within the interval?: You'll need to split the integral at the point of intersection and calculate the area on either side separately.
Can I use this calculator for functions of y with respect to x?: This calculator is designed for functions of x. For functions of y with respect to x, you would need to set up the integral with respect to y.
What if the curves don't intersect within the interval?: You can use the single integral formula without needing to split the interval.
How accurate is this calculator?: The calculator uses precise numerical integration methods to provide accurate results for most functions.
Can I use this for 3D surfaces?: No, this calculator is specifically for 2D curves in the xy-plane. For 3D surface areas, you would need a different tool.