1-12 - January 12, 2005 Today 5.4: Indefinite Integrals...

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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 .

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1-12 - January 12, 2005 Today 5.4: Indefinite Integrals...

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