Area between curves
MathDefinition
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
- 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
- Area between y = x² and y = x from x = 0 to x = 1 is ∫(x − x²)dx = 1/6
- 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
Study This Concept
Practice area between curves with free review games in these units: