AP Calculus BC
Advanced Placement Calculus BC including series, parametric equations, and polar functions.
Basic Concepts
Integration and Its Uses
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 \( \int v(t) dt \).
In a Nutshell
Integration helps us add up small pieces to find total amounts like area or distance.