6-3 Definite integrals & antiderivatives

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Presentation transcript:

6-3 Definite integrals & antiderivatives

Rules for Definite Integrals Order Zero Constant Multiple Sum & Difference

Rules for Definite Integrals Additivity (transitive-ish) Max-Min Inequality Domination Use 1 – 5 the most 

Ex 1) Suppose Find each of the following integrals, if possible. a) b) c) d) Not enough info

Ex 2) (Min-Max Inequality) Show that the value of Max value of 1

Def: Average / Mean Value The average / mean value of f (x) on [a, b] is Ex 3) Find the average value of f (x) = 4 – x2 on [0, 3]. Does f actually take on this value at some point in the given interval? Not a coincidence! Check out next Thm! fnInt Is f (x) = 1 anywhere in [0, 3]? 4 – x2 = 1 3 = x2 Yes! f (x) = 1 when

Thm: Mean Value Theorem At some point c on [a, b], *Let’s make some connections with what we’ve learned so far… (Note: F stands for any antiderivative) *deriv & integral are opposites * F is the antiderivative of f To find C, let x = a 0 = F(a) + C  –F(a) = C So…

Ex 4) Find using the formula

homework Pg. 286 # 36, 42, 43, 47, 48 Pg. 294 # 1, 5, 7, 9, 10, 15, 17, 20, 22, 24, 25, 26, 40, 47, 48