4-3 DEFINITE INTEGRALS MS. BATTAGLIA – AP CALCULUS.

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Presentation transcript:

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

EVALUATING A DEFINITE INTEGRAL

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 1.If f is defined at x = a, then we define 2.If f is integrable on [a,b], then we define

EVALUATING DEFINITE INTEGRALS

ADDITIVE INTERVAL PROPERTY If f is integrable on the three closed intervals determined by a, b, and c, then

USING THE ADDITIVE INVERSE PROPERTY

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 1.If f is integrable and nonnegative on the closed interval [a,b], then 2.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