Applications of Integration
Integration isn't just abstract math—it solves real problems in physics, engineering, economics, and everyday life.
1. Area Under a Curve
The most fundamental application. To find the area between f(x) and the x-axis from a to b:
$$A = \int_{a}^{b} |f(x)| \, dx$$
2. Physics Applications
Displacement from Velocity
$$s = \int v(t) \, dt$$
Work Done by a Force
$$W = \int_{a}^{b} F(x) \, dx$$
3. Volumes of Revolution
Disk Method
$$V = \pi \int_{a}^{b} [f(x)]^2 \, dx$$
Shell Method
$$V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx$$
4. Economics
- Consumer Surplus: Area between demand curve and market price
- Total Revenue: Integral of marginal revenue
- Present Value: Integral of discounted cash flows
5. Probability & Statistics
For continuous probability distributions, the probability that X falls in [a,b] is:
$$P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx$$