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MATH 151 Engineering Math I, Spring 2014 JD Kim Week15 Section 6.4 Section 6.4 The Fundamental Theorem of Calculus Ex1) Compute the following definite integrals using the graph given. 1-1) RA f (x) dx 1-2) RC f (x) dx 1-3) RB f (x) dx 1-4) RD f (x) dx 1-5) RA f (x) dx 1-6) RC f (x) dx 0 0 A 0 C C 1 The Fundamental Theorem of Calculus 1. Part I If f is continuous on [a, b], then the function g defined by Z x g(x) = f (t) dt a is continuous on [a, b] and differentiable on (a, b) and g ′ (x) = f (x). 2. Part II If f is continous on [a, b], then Z b a f (x) dx = F (b) − F (a), where F is an antiderivative of f . 2 Rx Ex1) If g(x) = 0 f (t)dt, where the graph of f (t) is given below, where 0 ≤ x ≤ 10, evaluate g(0), g(3), g(6) and g(10). Where is g(x) increasing? What is the maximum value of g(x)? 3 Ex2) Find the derivative Rx √ 2-1) g(x) = −1 t3 + 1 dt t2 dt t2 + 1 2-2) g(x) = R √x 2-3) g(x) = R sin x cos t dt x2 t 1 4 Ex3) Evaluate R2 1 3-1) 1 2 dx x 3-2) R0 3-3) R ln 6 3-4) R 2 x6 − x2 dx 1 x7 3-5) R2 0 (w 3 − 1)2 dw 3-6) R0 |x2 − 1| dx −1 (5x2 − 4x + 3) dx ln 3 −2 8et dt 5 Ex4) Evaluate R π/2 0 f (x) dx where f (x) = cos x if 0 ≤ x < π/4 sin x if π/4 ≤ x ≤ π/2 Ex5) Suppose an object is moving according to velocity v(t) = 3t − 5, 0 ≤ t ≤ 3. Find the displacement and distance traveled during the first 3 seconds. 6