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C Roettger, Spring 15 Math 165 – worksheet for ch. 5, Integration Chapter 5 covers topics including • Antiderivatives (OK, holdover from chapter 4, but resumed in 5.5) • Riemann Sums, Sigma Notation (5.2) • Simple Integrals, connection to Area (5.3) • FTC I (derivative of accumulation function is integrand) (5.4) • FTC II (evaluation theorem) (5.4) • Substitution Method (5.5) • Area between curves (5.6) Problem 1 Solve the initial value problem y = f (t) with y 0 = 2t + 5 cos(πt) y(2) = 18 Problem 2 a) Express the sum S = 1/1 + 1/2 + 1/3 + · · · + 1/100 using Sigma notation. b) Estimate the sum by comparing it to an integral over 1/x. Problem 3 Find the integral of the function f (t) graphed below between x = 0 and x = 13. Problem 4 Find the derivative of the following functions. Z z et dt, a) f (z) = 5 2 + sin t Z 4 1 b) g(u) = dt u t(1 + cos t) Z y 2 sin(t ) dt c) h(y) = sin 3 Z sin x t3 d) L(x) = dt. 2 cos x t + 1 4 Problem 5 a) Find the area under the graph of x3 e−x between x = 0 and x = 5. 4 b) Find the average value of x3 e−x between x = 0 and x = 5. Problem 6 If f (x) is continuous on [0, a], find the integral Z a f (x) M= dx. 0 f (x) + f (a − x) Try making the substitution u = a − x, and adding the resulting integral to M. Problem 7 Find the total area enclosed by the curves y = x3 − x + 4 y = x3 − x + x2