1. Area Under a Curve
The definite integral is used to calculate the area under a curve y = f(x) between two points a and b. This area is given by:
Area = ∫ab f(x) dx
If f(x) ≥ 0 in the interval [a, b], then the definite integral gives the area under the curve and above the x-axis.
2. Area Between Two Curves
When two functions f(x) and g(x) are defined on [a, b], and f(x) ≥ g(x), the area between them is:
Area = ∫ab [f(x) - g(x)] dx
This gives the vertical distance between the two curves, integrated over the interval.
3. Volume of Solids of Revolution
Volumes of solids formed by rotating a region around an axis can be calculated using definite integrals.
- Disk Method: If a function y = f(x) is rotated about the x-axis from x = a to x = b, the volume is:
V = π ∫ab [f(x)]² dx - Washer Method: If there's a hole (inner radius g(x)), the volume is:
V = π ∫ab ([f(x)]² - [g(x)]²) dx