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Unformatted text preview: SMU Classification: Restricted Part 2: Calculus B: Integral Calculus 1.Integration Techniques 2.Applications Version: 1.0 Date: Oct 20, 2018 Textbook, Chapter 5, Chapter 6 Grace in Mathematics Associate Professor Christopher Ting Page 1 of 19 SMU Classification: Restricted Version: 1.0 Date: Sep 20, 2018 Grace in Mathematics Associate Professor Christopher Ting Page 2 of 19 SMU Classification: Restricted Definite Integral indefinite integral : antiderivative ? definite integral Grace in Mathematics Associate Professor Christopher Ting Page 3 of 19 SMU Classification: Restricted Definite Integral and Area Between Curves Textbook, p. 371 4 left-rectangles Grace in Mathematics 4 right-rectangles Associate Professor Christopher Ting Page 4 of 19 SMU Classification: Restricted lim ෍ ∆ → න →∞ =1 Textbook, p. 375, THEOREM 3 Grace in Mathematics Associate Professor Christopher Ting Page 5 of 19 SMU Classification: Restricted Derivative is a limit. Definite integral is also a limit. Grace in Mathematics Associate Professor Christopher Ting Page 6 of 19 SMU Classification: Restricted The Fundamental Theorem of Calculus FC to F(a) Change h 1. න → + → 2. න = − () IMPORTANT: Computation of definite integrals without finding the antiderivatives will get penalty, because your calculator may do the computation for you. Textbook, p. 383 Grace in Mathematics Associate Professor Christopher Ting Page 7 of 19 Textbook, p. 377 SMU Classification: Restricted ⇔ − =0 ⇔ − = −[ − ] ⇔ − = [ − ] ⇔ ± − ± = − ± − ⇔ − = − Grace in Mathematics + [ − ] Associate Professor Christopher Ting Page 8 of 19 SMU Classification: Restricted In-Class Exercise Q142 න 2 + 3 4 − = 2 + 3 − 4 ln + 2 න 1 2 + 3 Log rule 4 − = 22 + 3 2 − 4 ln 2 − 12 + 31 − 4 ln 1 Textbook, p. 383, EXAMPLE 2 = 3 + 3 2 − 3 − 4 ln 2 Grace in Mathematics Associate Professor Christopher Ting Page 9 of 19 Textbook, p. 384, EXAMPLE 3 SMU Classification: Restricted In-Class Exercise Q143 integration by substitution Grace in Mathematics Associate Professor Christopher Ting Page 10 of 19 Textbook, p. 384 SMU Classification: Restricted integration by substitution Evaluate with the range of u Grace in Mathematics Associate Professor Christopher Ting Page 11 of 19 SMU Classification: Restricted Area between Curves Textbook, p. 400 + =? Grace in Mathematics Associate Professor Christopher Ting Page 12 of 19 SMU Classification: Restricted Area between Curves Q: How do we know where ≥ ()? Textbook, p. 400, THEOREM 1 Grace in Mathematics Associate Professor Christopher Ting Page 13 of 19 SMU Classification: Restricted Let = 0. Q: Is ≥ () over the interval [, ]? Textbook, p. 401, EXAMPLE 1 Grace in Mathematics A: I don’t know. How about drawing a sign chart? Associate Professor Christopher Ting Page 14 of 19 SMU Classification: Restricted Drawing the sign chart of − () 2 = 6 − , = 0 → − = 6 − + − 0 [, ] 2 − 6 Ans: ≥ () over [1, 4] Grace in Mathematics Associate Professor Christopher Ting Page 15 of 19 SMU Classification: Restricted In-Class Exercise Q144 SOLUTION: 4 4 = න [ − ] = න 6 − 2 = ⋯ = 24 Textbook, p. 401 1 1 Grace in Mathematics Associate Professor Christopher Ting Page 16 of 19 SMU Classification: Restricted Drawing the sign chart of − () = 2 − 2, = 0 → − = 2 − 2 − + −1 Textbook, p. 401, EXAMPLE 2 0 1 Grace in Mathematics = x(x-2) + 2 Associate Professor Christopher Ting Page 17 of 19 SMU Classification: Restricted In-Class Exercise Q145 =x(2-x) 1. Create the sign chat when f(x) = 0, what is x as that is when graph-wise the value changes from +ve to -ve & vice versa 2 2 = න [ − ] = ⋯ = 3 1 0 1 4 2 = න [ − ] + න [ − ] = ⋯ = + = 2 3 3 −1 0 Textbook, p. 401 Grace in Mathematics Associate Professor Christopher Ting Page 18 of 19 Takeaways SMU Classification: Restricted Grace in Mathematics Associate Professor Christopher Ting Page 19 of 19 ...
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