AP Calculus BC

Advanced Placement Calculus BC including series, parametric equations, and polar functions.

Limits and ContinuityDifferentiation and ApplicationsIntegration and Its Uses
Infinite Series and ConvergenceParametric Equations and MotionPolar Coordinates and Graphs
Analyzing Population GrowthDesigning Roller Coasters with Parametric EquationsSignal Processing and Polar Graphs
Mastering Multiple-Choice QuestionsEffective Free-Response Techniques
Limits and Continuit...Differentiation and ...Integration and Its ...Infinite Series and ...Parametric Equations...Polar Coordinates an...Analyzing Population...Designing Roller Coa...Signal Processing an...Mastering Multiple-C...Effective Free-Respo...
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.

Next
Infinite Series and Convergence
Integration and Its Uses - AP Calculus BC Content | Practice Hub