Area Between Curves Integral Calculator

What is the area between curves?

The area between two curves is the region enclosed by two functions over a specific interval. Calculating this area requires integrating the difference between the upper and lower functions over the interval of interest.

This concept is fundamental in calculus and has applications in physics, engineering, and economics. Understanding how to compute the area between curves helps in solving problems involving accumulation, such as finding the volume of revolution or the work done by a variable force.

How to calculate the area between curves

To calculate the area between two curves, follow these steps:

1. Identify the upper and lower functions over the interval [a, b].
2. Find the points of intersection of the two curves to determine the interval(s) of integration.
3. Set up the integral as the difference between the upper and lower functions.
4. Evaluate the integral to find the area.

This process ensures that you accurately capture the region between the curves and compute its area.

Formula

Where:

• f(x) is the upper function
• g(x) is the lower function
• [a, b] is the interval of integration

This formula assumes that f(x) ≥ g(x) over the interval [a, b]. If this condition is not met, you may need to adjust the limits of integration or split the integral into multiple parts.

Example calculation

Let's calculate the area between the curves y = x² and y = x from x = 0 to x = 1.

1. Identify the upper and lower functions: f(x) = x² (upper), g(x) = x (lower).
2. Set up the integral: ∫[1 to 0] (x² - x) dx.
3. Evaluate the integral:
   • ∫x² dx = (x³)/3
   • ∫x dx = (x²)/2
   • Combine the results: [(1³)/3 - (1²)/2] - [(0³)/3 - (0²)/2] = (1/3 - 1/2) - (0 - 0) = -1/6
4. Take the absolute value of the result to get the area: 1/6.

The area between the curves is 1/6 square units.