Math
Area between curves.
Definition
The area of the region enclosed between two curves f(x) and g(x) is found by integrating the absolute difference of the functions over the interval where they bound a region. You integrate (top function − bottom function) with respect to x.
How it works · 5 phases
Step by step.
- Find the points of intersection by setting f(x) = g(x).
- Determine which function is on top in each subinterval.
- Set up the integral ∫ from a to b of [f(x) − g(x)] dx where f(x) ≥ g(x).
- Evaluate the definite integral.
- If the curves switch which is on top, split into separate integrals.
Examples
Real-world.
- 1 Area between y = x² and y = x from x = 0 to x = 1 is ∫(x − x²)dx = 1/6
- 2 Finding the area enclosed between y = sin(x) and y = cos(x) from 0 to π/2
Key Fact
A = ∫ from a to b |f(x) − g(x)| dx