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Section 4.2 Definite Integral Math 1231: Single-Variable Calculus Slide 2 Definite Integral: Definition Slide 3 Slide 4 Slide 5 Net Area (Signed Area) Slide 6 Existence of Definite Integral Theorem If f is continuous on [a, b], or if f has only a finite number of jump discontinuities, then f is integrable on [a, b]; that is, the definite integral exists. Slide 7 Evaluate Integrals How to evaluate integrals? 1.Interpret the integrals in terms of areas. 2.Fundamental Theorem of Calculus. 3.Other techniques. Slide 8 Examples Example Evaluate the following integral by interpreting it in term of (signed) areas. The integral is equal to the area of blue region, which is a quarter circle of radius 1. Slide 9 Properties of the Integral Slide 10 Slide 11 Examples Example Slide 12 Evan and Odd Functions Slide 13 More Examples Slide 14 Slide 15 Slide 16 Slide 17 Slide 18 Slide 19