Antiderivatives and indefinite integration2009

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Antiderivatives and indefinite integration2009

  1. 1. Drive activity • Please submit your drive activities.
  2. 2. Sec. 4.1: Antiderivatives
  3. 3. Uses of integration 1. 2. 3. 4. 5. “Undoes” differentiation. Finds the area under a curve. Finds the volume of a solid. Finds the center of mass. Finds s(t) given a(t) or v(t). *** and many more….
  4. 4. The Uniqueness of Antiderivatives 2 Suppose f x 3x , find an antiderivative of f. That is, 2 find a function F(x) such that F ' x 3x . F x x 3 Using the Power Rule in Reverse Is this the only function whose derivative is 3x2? H x H' x x 3 5 3x 2 K x x 3 11 3x 2 K' x M x M' x x 3 3x 2 There are infinite functions whose derivative is 3x2 whose general form is: C is a constant real G x 3 x C number (parameter)
  5. 5. When we find antiderivatives and add the constant C, we are creating a family of curves for each value of C. x3 C C=0 C=1 C=3
  6. 6. Indefinite Integral f ( x)dx Integral Sign Integrand F ( x) C The Indefinite Integral Variable of Integration The constant of Integration
  7. 7. Indefinite Integral The indefinite integral gives a family of functions! (Not a value) The indefinite integral always has a constant! Vs. The definite integral (later) gives a numerical value.
  8. 8. Summary Integration is the “inverse” of differentiation. d F ( x) dx dx F ( x) C Differentiation is the “inverse “ of integration. d dx f ( x)dx f ( x)
  9. 9. Basic Integration Rules 0dx C kdx kx C k f ( x )dx k f ( x )dx
  10. 10. Basic Integration Rules f ( x ) g ( x ) dx f ( x )dx n 1 n x dx x C n 1 Power Rule g ( x )dx
  11. 11. Basic Integration Rules cos xdx sin xdx 2 sec xdx sin x C cos x C tan x C
  12. 12. Basic Integration Rules 2 csc xdx sec x tan xdx csc x cot xdx cot x C sec x C csc x C
  13. 13. Examples Find each of the following indefinite integrals. 5 a. x dx b. 5 x3 dx x sin x dx c. 1 6 d. 1 x 6 C cos x C 5 4 x4 C dx 2 x C
  14. 14. Application problem • A ball is thrown upward with an initial velocity of 64 ft/ sec from an initial height of 80 feet. a. Find the position function, s(t) b. When does the ball hit the ground?
  15. 15. Example A particle moves along a coordinate axis in such a way that 3 its acceleration is modeled by a t 2t for time t > 0. If the particle is at s = 5 when t = 1 and has velocity v = - 2 at this time, where is it when t = 4? Integrate the acceleration to find velocity: v t 2t 3 dt 2 t 3 1 3 1 2 dt t 3 1 t 2 t C 2 C Use the Initial Condition to find C for velocity: 2 1 2 C C v t 1 1 Integrate the Velocity to find position: s t t 5 1 2 1 Answer the Question: 1 dt t 2 dt 1 2 1 1 dt t 2 1 1t C t Use the Initial Condition to find C for position: 1 C s 4 C 4 1 4 5 5 1.25 s t t 1 1 t C t 5

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