4/6/2010 Integrals Integrals MAC 2233 Antiderivatives A function - PDF document

4/6/2010 Integrals Integrals MAC 2233 Antiderivatives A function F is an antiderivative of f on an interval I if _______________ for every x in I . If G is an antiderivative of f , then every antiderivative of f must have the form where C is ___________ Indefinite Integrals The process of finding all antiderivatives of a function is called antidifferentiation or integration 1

4/6/2010 Rules of Integration ∫ = + , where , are constants k dx kx C k C 1 1 ∫ ∫ = + + ≠ − n n 1 , where 1 x dx x C n + 1 n ∫ ∫ = ( ) ( ) cf x dx c f x dx ∫ ∫ ∫ ± = ± [ ( ) ( )] ( ) ( ) f x g x dx f x dx g x dx Rules of Integration 1 ∫ = + ln | | dx x C x ∫ = + x x e dx e C Example ∫ − 4 dx • Integrate 2

4/6/2010 Example ∫ + − 2 1 3 6 x x dx • Integrate Example ∫ − − 1.7 2.5 x x dx • Integrate Example 1 ∫ + − 1 2 x dx • Integrate 3 x • Rewrite • Rewrite 3

4/6/2010 Example ∫ 5 9 x dx • Integrate • Rewrite • Rewrite Example 0.4 4.2 x ∫ + − x 2 e dx • Integrate 0.4 3 x • Rewrite • Rewrite Homework • p. 381 problems 1-29 odd, 37, 43, 45 4

4/6/2010 S ubstitution ∫ + 2 3 7 3 ( 1) How do we integrate x x dx ? 1. Let u = g ( x ), where g is part of the integrand, usually the _______________ of the composite usually the of the composite function f ( g ( x )). 2. Compute ______________ . 3. Use the substitution __________________ to convert the entire integral into one involving only u . 4. Evaluate the resulting integral. 5. Replace u by g ( x ) to obtain the final solution as a function of x . Example 3 ∫ + 2 3 3 ( 2) 2 • Integrate t t dt Example + 2 3 x 2 ∫ • Integrate dx + 3 2 ( x 2 ) x 5

4/6/2010 Example • Integrate ∫ + 3 4 7 x ( x 9 ) dx Example x ∫ dx • Integrate x + 3 2 Example 1 ∫ dx • Integrate − 7 5 x 6

4/6/2010 Example ∫ 2 x xe dx • Integrate Example The current circulation of the Investor’s Digest is 3000 copies per week. The managing editor of the weekly projects a growth rate of weekly projects a growth rate of copies per week, t weeks from now, for the next 3 years. Based on her projection, what will the circulation of the digest be 125 weeks from now? From Calculus for the Managerial, Life, and Social Sciences, 6 th ed. By Tan, 2003, example 12, p.406. Homework • p. 394 problems 3-35 odd, 45, 51, 55, 61, 67 7

4/6/2010 Area Under the Curve How do we calculate the area of the region bounded by the graph of a nonnegative function, f , the x -axis, and the vertical lines x = a and x = b ? Area under the curve Let f be a nonnegative, continuous function on [ a, b ]. Then the area of the region under the graph of f is graph of f is The Definite Integral Let f be a continuous function defined on [ a , b ]. If exists for all choices of x 1 , … , x n in the subintervals of [ a , b ] then this limit is called the definite integral of f from a to b and we write 8

4/6/2010 Properties of the Definite Integral b b ∫ ∫ = ( ) ( ) kf x dx k f x dx a a b b b ∫ ∫ ∫ ∫ ∫ ∫ ± ± = ± ± [ ( ) [ ( ) f x f x g x g x ( )] ( )] dx dx f x dx f x dx ( ) ( ) g x dx g x dx ( ) ( ) a a a b c b ∫ ∫ ∫ = + ( ) ( ) ( ) f x dx f x dx f x dx a a c a ∫ f x dx = ( ) 0 a The Fundamental Theorem of Calculus Let f be a continuous function on [ a , b ]. Then where F is any antiderivative of f ; that is F ’( x ) = f ( x ). We write Example • Find the area of the region under f ( x ) = 4 x – 1 on the interval [2, 4]. 9

4/6/2010 Example 0 ∫ − 4 x dx • Evaluate − 1 Example 2 ∫ − + 5 3 1 • Evaluate t t dt 1 Example 1 ∫ − 2 2 • Evaluate 3 ( 1) x x dx 0 10

4/6/2010 Example 0 3 t ∫ dt • Evaluate − 4 4 1 (2 t ) − Example 0 1 ∫ dx • Evaluate − 4 5 x − 1 Net Change • The definite integral represents the net change in the antiderivative function 11

4/6/2010 Example A certain oil well that yields 400 barrels of crude oil a month will run dry in 2 years. The price of crude oil is currently $95 per barrel and is crude oil is currently $95 per barrel and is expected to rise at a constant rate of 30 cents per barrel per month. If the oil is sold as soon as it is extracted from the ground, what will be the total future revenue from the well? From Calculus for Business, Econom ics and the Social and Life Sciences, 10 th ed. By Hoffmann & Bradley, 2007, problem 50, p.412. Homework • p. 410 problems 1-29 odd, 41, 43, 45, 49, 53, 55, 57 59 63 57, 59, 63 12