Study Notes
Integration is the reverse process of differentiation, used to find the original function from its derivative. It involves adding an arbitrary constant, C, to account for the constant lost during differentiation.
- Integration — the process of finding the original function from its derivative. Example: If f'(x) = 2x, then f(x) = x² + C.
- Indefinite Integration — integration without specific limits, resulting in a general form with a constant. Example: ∫x dx = 0.5x² + C.
- Definite Integration — integration with specific limits, used to find the area under a curve. Example: ∫ from a to b of x dx = [0.5x²] from a to b.
- Volume of Revolution — the volume of a 3D shape formed by rotating a region around an axis. Example: Rotating y = x² around the x-axis.
Exam Tips
Key Definitions to Remember
- Integration is the reverse of differentiation.
- Indefinite integration includes a constant C.
- Definite integration calculates the area under a curve.
Common Confusions
- Forgetting to add the constant C in indefinite integration.
- Mixing up definite and indefinite integration.
Typical Exam Questions
- What is the integral of x²? Answer: (1/3)x³ + C
- How do you find the area under the curve y = x from x = 1 to x = 3? Answer: Evaluate ∫ from 1 to 3 of x dx.
- How do you find the volume of revolution for y = x² about the x-axis? Answer: Use the formula π∫ from a to b of (f(x))² dx.
What Examiners Usually Test
- Understanding of integration as the reverse of differentiation.
- Ability to solve problems involving definite and indefinite integrals.
- Application of integration to find areas and volumes.