Integrals

The integral is the inverse operation of differentiation, representing the accumulation of quantities.

Definition

The definite integral of a function from to is defined as:

This represents the area under the curve from to .

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration:

This theorem shows that differentiation and integration are inverse operations.

Common Integrals

Here are some common integrals:

1. Power rule: (for )
2. Exponential:
3. Trigonometric:

Applications

Integrals have numerous applications in:

• Calculating areas and volumes
• Physics (work, energy)
• Probability (probability distributions)
• Economics (total cost, total revenue)