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ReviewC5 - Sections 5.1 5.2 5.3 5.4 and 5.5 Review Theory...

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Sections 5.1, 5.2, 5.3, 5.4 and 5.5: Review Theory Instructor: Ms. Hoa Nguyen ([email protected]) 5.1 The Area Problem : Find the area of the region S that lies under the curve y = f ( x ) from a to b . First, we approximate the region S by n rectangles (using left endpoints, right endpoints or mid- points x * i ). Then we take the limit of the sum of the areas of these rectangles (Riemann sum) as we increase the number of rectangles. So, define the actual area A of the region S to be the limit of the Riemann sum: A = lim n →∞ Σ n i =1 f ( x * i ) x = b a f ( x ) dx 5.2 Some Important Properties of the Definite Integral b a f ( x ) dx = - a b f ( x ) dx a a f ( x ) dx = 0 b a cf ( x ) dx = c b a f ( x ) dx where c is any constant. b a [ f ( x ) ± g ( x )] dx = b a f ( x ) dx ± b a g ( x ) dx c a f ( x ) dx + b c f ( x ) dx = b a f ( x ) dx If f ( x ) 0 for a x b , then b a f ( x ) dx 0. 5.3 The Fundamental Theorem of Calculus Suppose f is a continuous function on [ a, b ], 1. If g ( x ) = x a f ( t ) dt , then g ( x ) = d dx x a f ( t ) dt = f ( x ).
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