Integration: The Art of Accumulation
If differentiation is about slicing things up to find instantaneous rates, Integration is about gluing them back together to find totals. It is the mathematical tool used to calculate areas, volumes, and central points of irregular shapes.
1. Indefinite Integrals (The Antiderivative)
An indefinite integral asks: "What function, when differentiated, gives me this result?" It represents a family of functions.
Why "+ C"? because the derivative of a constant is zero. If you differentiate x² + 5, you get 2x. If you differentiate x² + 100, you also get 2x. When we go backwards (integrate 2x), we don't know which constant was there originally, so we just write "+ C".
2. Definite Integrals (Area Under Curve)
A definite integral has start and end points [a, b]. It calculates the exact Net Signed Area between the function curve and the x-axis.
Area accumulation from x=a to x=b
Plug limits into the antiderivative
3. How to Calculate It
- Power Rule for Integration: Add 1 to the exponent, then divide by the new exponent.
- Example: ∫ x² dx = x³/3 + C
- Simpson's Rule: For functions that can't be integrated simply, computers (like this calculator) use numerical methods like Simpson's Rule to approximate the area with high precision.
4. Real World Applications
- Physics: Integrating velocity over time gives you **Distance**. Integrating Force over distance gives you **Work**.
- Economics: Integrating a "Marginal Cost" function gives you the **Total Cost** of production.
- Probability: To find the probability of an event happening within a range (e.g., getting between 80-90 on a test), statisticians integrate the Probability Density Function (Bell Curve).