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Page 1 Math 151-copyright Joe Kahlig, 13C Sections 6.4: The Fundamental Theorem of Calculus Definition: Z The indefinite integral of f is used to indicate the process of finding the antiderivative of f . i.e. f (x) dx = F (x) + C Example: Compute. A) Z 2x5 + 7x + 4 dx = B) Z 3x2 da = The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a, b], then Zb a f (x) dx = F (b) − F (a), where F is any antiderivative of f . Example: Compute the following. A) Z5 3x2 + 4x + 2 dx = 1 B) Z4 0 3x + 8e4x dx = Math 151-copyright Joe Kahlig, 13C C) Z5 1 dx = x2 −2 D) Z3 0 E) |x2 − 4| dx = Z9 √ 4 1 x− √ x 2 dx = Page 2 Math 151-copyright Joe Kahlig, 13C Page 3 The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then the function g defined by g(x) = Zx f (t) dt, a with a ≤ x ≤ b is continuous on [a, b] and is differentiable on (a, b), and g′ (x) = f (x) Example: Find g′ (x). A) g(x) = Zx t2 + 1 dt Zx2 tan3 (t) dt a B) g(x) = 4 C) g(x) = Z2 ln(u) du x3 D) g(x) = xZ3 +1 x2 u5 + 2 du Page 4 Math 151-copyright Joe Kahlig, 13C Example: Use the graph of f ′ (x) to answer these questions. 6 4 2 20 5 −2 10 25 30 15 −4 A) Which is larger? f (15) or f (25) B) If f (5) = 30, find f (20). Example: Define g(a) by g(a) = Za f (x) dx where f (x) is the graph given below. Find the intervals 0 where g(a) is increasing. If possible, give the values of the absolute maximum and absolute minimum. 6 4 2 15 −2 −4 5 10 20 25 30 35 40 Math 151-copyright Joe Kahlig, 13C Page 5 Example: A particle is moving in straight line motion that is expressed by the formula: v(t) = t2 −t−6 (measured in meters per second). A) Find the displacement from t = 1 to t = 4. B) Find the total distance traveled from t = 1 to t = 4.