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4-3 definite integrals. Ms. Battaglia – ap calculus . Definite integral. A definite integral is an integral with upper and lower bounds. The number a is the lower limit of integration, and the number b is the upper limit of integration.
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4-3 definite integrals Ms. Battaglia – ap calculus
Definite integral • A definite integral is an integral • with upper and lower bounds. The number a is the lower limit of integration, and the number b is the upper limit of integration.
Theorem 4.4 (Continuity implies Integrability) • If a function f is continuous on the closed interval [a,b], then f is integrable on [a,b].
The first fundamental theorem of calculus • If f is continuous on the closed interval [a,b] and F is the indefinite integral of f on [a,b], then
Areas of common Geometric Figures • Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula. • a. b. c.
Definition of two Special integrals • If f is defined at x = a, then we define • If f is integrable on [a,b], then we define
Additive Interval Property • If f is integrable on the three closed intervals determined by a, b, and c, then
Properties of Definite Integrals • If f and g are integrable on [a,b] and k is a constant, then the function of kf and f + g are integrable on [a,b], and • 1. • 2.
Evaluation of a definite integral • Evaluate using each of the following values.
Preservation of Inequality • If f is integrable and nonnegative on the closed interval [a,b], then • If f and g are integrable on the closed interval [a,b] and f(x) < g(x) for every x in [a,b], then
Homework • Page 278 #9, 11, 18, 31, 42, 43, 47, 49, 65-70