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Math 151 Section 6.4 The Fundamental Theorem of Calculus Indefinite Integral 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: Evaluate the following integrals. A. ! 2x B. ! 3x 5 + 7x + 4 dx 2 da The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then b " f ( x) dx = F (b) ! F (a) where F is any antiderivative of f. a Example: Evaluate the following integrals. 5 A. ! 3x 2 + 4x + 2 dx 1 4 B. ! 3x + 8e 0 5 C. ! 3x 1 2 da 4x dx Math 151 ! 4 D. ! sec( x) tan( x) dx 0 5 E. " !2 1 dx x2 3 F. " x 2 ! 4 dx 0 The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a, b], then the function g x defined by g ( x) = ! f (t ) dt with a ! x ! b is continuous on [a, b], is differentiable on (a, b), and a g !( x) = f ( x) . Example: Find g !( x) . x A. g ( x) = ! t 2 +1dt a x2 B. g ( x) = ! t 3 + 2 dt 4 Math 151 2 C. g ( x) = ! ln (u) du x3 D. g ( x) = x 3 +1 ! u5 + 2 du x2 Example: Use the graph of f !( x) , shown below, to answer the following questions. A. Which is greater, f (15) or f (25)? B. If f (5) = 30, what is f (0)? C. If f (5) = 30, what is f (20)? Math 151 a Example: Compute g(10) where g(a) is defined by g ( a) = ! 2 f ( x) dx and f (x) is the graph shown 0 below. Example: A particle is moving in straight line motion and its velocity is described by the function v (t ) = t 2 !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.