Study Notes
Integration involves finding the area under curves, calculating volumes of revolution, and solving differential equations using various techniques.
- Area under curve defined parametrically — Calculating the area using parametric equations. Example: Find the area under a curve given in parametric form.
- Volume of revolution around the x-axis — Volume formed by rotating a curve around the x-axis. Example: Calculate the volume when y = f(x) is rotated around the x-axis.
- Integration by substitution — Simplifying integrals by changing variables. Example: Use substitution to integrate a complex function.
- Integration by parts — Technique used when integrals involve products of functions. Example: Integrate x * e^x using integration by parts.
- Partial fractions — Decomposing rational functions to simplify integration. Example: Integrate (x^2 + 3x + 2)/(x + 1)(x + 2) using partial fractions.
- Solving differential equations — Finding functions that satisfy given differential equations. Example: Solve dy/dx = 3y for y.
- Modelling with differential equations — Using differential equations to model real-world scenarios. Example: Model population growth with dy/dt = ky.
Exam Tips
Key Definitions to Remember
- Integration by substitution
- Integration by parts
- Partial fractions
- Differential equations
Common Confusions
- Mixing up integration by parts and substitution
- Incorrectly setting up partial fractions
- Misinterpreting differential equations
Typical Exam Questions
- How do you find the volume of revolution around the x-axis? Use the formula for volume of revolution.
- What is the method for integrating a product of functions? Use integration by parts.
- How do you solve a differential equation? Separate variables and integrate both sides.
What Examiners Usually Test
- Ability to apply integration techniques correctly
- Understanding of when to use each integration method
- Solving and interpreting differential equations