Summary
Integration is the process of finding a function from its derivative, known as the anti-derivative or integral. It is the reverse of differentiation and involves adding a constant of integration. Integration can be used to find areas under curves, volumes of revolution, and solve improper integrals.
- Integration as the Reverse of Differentiation — Integration finds the original function from its derivative. Example: If f'(x) = 2x, then f(x) = x² + C.
- Constant of Integration — An arbitrary constant added to the integral of a function. Example: If the integral of f'(x) is x², then f(x) = x² + C.
- Definite Integration — Calculates the area under a curve between two points. Example: ∫ from a to b of f(x) dx gives the area under f(x) from x = a to x = b.
- Improper Integrals — Integrals with infinite limits or undefined points. Example: ∫ from 1 to ∞ of 1/x² dx.
- Volume of Revolution — Volume formed by rotating a curve around an axis. Example: Rotating y = x² around the x-axis from x = 0 to x = 1.
Exam Tips
Key Definitions to Remember
- Integration is the reverse of differentiation.
- Constant of integration is the arbitrary constant added to indefinite integrals.
- Definite integrals calculate the area under a curve between two limits.
- Improper integrals involve infinite limits or undefined points.
- Volume of revolution is the volume formed by rotating a curve around an axis.
Common Confusions
- Forgetting to add the constant of integration in indefinite integrals.
- Mixing up definite and indefinite integrals.
- Misunderstanding the limits in improper integrals.
Typical Exam Questions
- What is the integral of 2x? x² + C
- How do you find the area under y = x² from x = 0 to x = 2? Evaluate ∫ from 0 to 2 of x² dx
- What is the volume when y = x² is rotated around the x-axis from x = 0 to x = 1? Use the formula for volume of revolution.
What Examiners Usually Test
- Understanding of integration as the reverse of differentiation.
- Ability to evaluate definite and improper integrals.
- Application of integration to find areas and volumes.