Integral Area Between Two Curves Calculator

What is the area between two curves?

The area between two curves is the region enclosed by two functions over a specific interval on the x-axis. This concept is fundamental in calculus and has applications in physics, engineering, and economics.

To find this area, you need to determine where the two curves intersect and then calculate the integral of the difference between the upper and lower functions over the interval between the points of intersection.

How to calculate the area between two curves

Calculating the area between two curves involves several steps:

1. Identify the upper and lower functions over the interval.
2. Find the points of intersection between the two curves.
3. Set up the integral with the difference between the upper and lower functions.
4. Evaluate the integral between the points of intersection.

For example, if you have the functions f(x) = x² and g(x) = x + 2, and they intersect at x = -1 and x = 2, the area between them would be calculated as:

Example calculation

Let's calculate the area between the curves y = x² and y = x + 2 from x = -1 to x = 2.

1. First, find the points of intersection by setting the functions equal to each other:

   Intersection Points

   x² = x + 2

   x² - x - 2 = 0

   Solutions: x = -1 and x = 2

2. Determine which function is upper and which is lower over the interval [-1, 2].
3. Set up the integral:

   Integral Setup

   Area = ∫[-1 to 2] [(x + 2) - (x²)] dx

4. Calculate the integral:

   Integral Calculation

   ∫(x + 2 - x²) dx = (x²/2 + 2x - x³/3) evaluated from -1 to 2

   At x = 2: (4/2 + 4 - 8/3) = 2 + 4 - 2.666... ≈ 3.333

   At x = -1: (1/2 - 2 + (-1)/3) = 0.5 - 2 - 0.333... ≈ -1.833

   Area = 3.333 - (-1.833) ≈ 5.166

The area between the curves is approximately 5.166 square units.