Basic Concepts
In a nutshell: Integration helps us add up small pieces to find total amounts like area or distance.
## The Area Under the Curve
Integration is the reverse of differentiation. It helps us find areas, volumes, and accumulated values.
- The definite integral \( \int_a^b f(x) dx \) gives the area under \( f(x) \) from \( x = a \) to \( x = b \).
- The Fundamental Theorem of Calculus connects derivatives and integrals.
## Applications
- Calculating area between curves
- Finding total distance traveled
- Computing work done by a force
## Making Sense of the World
Integrals appear in physics (energy, work), economics (total cost), and biology (population growth).
Examples
- The area under \( y = x \) from 0 to 2 is \( \int_0^2 x dx = 2 \).
- Total distance from a velocity function using \( \int v(t) dt \).