Definite Integration Cheat Sheet

Key Definitions

• Definite Integral: Represents the area under the curve of a function between two points. ✅

• Limits of Integration: The two bounds (a and b) in the integral notation. ✅

• Fundamental Theorem of Calculus: Connects differentiation and integration, stating that if F is an antiderivative of f, then:

∫[a to b] f(x) dx = F(b) - F(a) ✅

Important Concepts

Properties of Definite Integrals

• Linearity:

∫[a to b] [c f(x) + g(x)] dx = c ∫[a to b] f(x) dx + ∫[a to b] g(x) dx ✅

• Additivity:

∫[a to c] f(x) dx + ∫[c to b] f(x) dx = ∫[a to b] f(x) dx ✅

• Reversal of Limits:

∫[a to b] f(x) dx = -∫[b to a] f(x) dx ✅

Key Formulas

• Definite Integral:

∫[a to b] f(x) dx = F(b) - F(a) ✅

• Area under the curve:

A = ∫[a to b] f(x) dx ✅

Common Mistakes to Avoid

Common Mistake: Forgetting to apply the limits of integration correctly when evaluating the definite integral.

Memory Tricks

Tip: "FIND" —

• F: Function to integrate

• I: Identify limits

• N: Note antiderivative

• D: Determine area (evaluate at limits)

Key Diagrams

Area under curve:
          |
          |         *
          |        | 
          |        |   
          |        |     
          |        |       
          |---------------------
          a         b