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Unformatted text preview: January 12, 2005 Today 5.4: Indefinite Integrals 5.4: Net Change Theorem and applications 1 Definition. Let f be a continuous function on an interval. The indefinite integral of f is the most general antiderivative of f and it is denoted by f (x)dx. In particular the notation F (x) = f (x)dx means F (x) = f (x). Notation: Let f be a continuous function on the interval [a, b]. b a x a f (x) dx f (t) dt is a number is an antiderivative of f whose value at a is 0 f (x) dx is the most general antiderivative of f Let f be a continuous function on [a, b]. Let F be an antiderivative of f . b a x a f (x) dx = F (b)  F (a) f (t) dt = F (x)  F (a) f (x) dx = F (x) + C Table of indefinite integrals xn+1 xn dx = + C (n = 1) n+1 1 dx = ln x + C x ex dx = ex + C sin x dx =  cos x + C cos x dx = sin x + C sec2 x dx = tan x + C sin x dx = sec x + C 2x cos 1 dx = arctan x + C 2 1+x 1 1  x2 dx = arcsin x + C A honeybee population starts with 100 bees and increases at a rate of n (t) bees per week. What does 100 + represent?
15 0 n (t) dt The net change rule The integral of the rate of change is the net change,
b a F (t) dt = F (b)  F (a). If an object moves along a straight line with position s(t) its velocity is v(t) = s (t) and
t2 t1 v(t) dt = s(t2)  s(t1) = net change of position = displacement t2 t1 v(t) dt = total distance traveled Displacement vs total distance traveled Suppose v(t) = 3t  5 on 0 t 3. (a) Find the displacement (b) Find the total distance traveled between t = 0 and t = 3. ...
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This note was uploaded on 03/09/2008 for the course CALC 1,2,3 taught by Professor Varies during the Spring '08 term at Lehigh University .
 Spring '08
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 Calculus, Definite Integrals, Derivative, Integrals

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