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# Math 122 Final Study Guide Math 122

Drexel

GPA 3.72

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This 7 page Study Guide was uploaded by Ji-woo Kim on Tuesday January 19, 2016. The Study Guide belongs to Math 122 at Drexel University taught by H. Lee in Summer 2015. Since its upload, it has received 22 views. For similar materials see Calculus II in Mathematics (M) at Drexel University.

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Date Created: 01/19/16

Ji-woo Kim 14072463 Math 113 Jan 25, 2016 Math 122 Midterm 2 Study Guide (Chapters: 6.2, 6.4, 6.6, 7.2, 7.8, 8.2, 7.3) Chapter 6.2: Volume ▯ ∗ ???????????????????????? ≈ ????(???? ▯∆???? ▯ ▯▯▯ ▯ = lim ????(???? )∆???? ▯→▯ ▯ ▯ ▯▯▯ ▯ = ???? ???? ???????? ▯ * A(x) = area of cross section ** Revolving about a horizontal line y = l Solids of Revolution : Solids generated by revolving plane region about a fixed line (axis of revolution) : ???? = ????????(????) ▯ ▯ ▯ ▯ ???? = ???? ????(????) − ???? ???? ???????? ▯ ▯ ???? = ????(????) ???????? ▯ * Revolving about a vertical line x=m Solids not form revolving whose cross section is not circular but cross section is known ▯ Ex) ???? = ???? A x = ???? ∗ ???? = ???? ▯ ▯ ???? = ▯ ???? ???? ???????? Chapter 6.4: Length of a Plane Curve Total Length ≈ ▯ ????▯ ▯▯▯ ▯ ▯ ▯ = (∆???? ▯ +(∆???? ) ▯ ▯▯▯ Ji-woo Kim 14072463 Math 113 Jan 25, 2016 ▯ ▯ ▯ = ▯→▯ (????????) +(????????) ▯▯▯ ▯ ▯ ▯ = (????????) +(????????) ▯ ▯ ???????? ▯ = ???????? ▯ 1 + ???????? ▯ ▯ ▯ ???????? = 1 + ???????? = ???? ▯ ???????? Chapter 6.6: Work W = F*d 3 Work Problems 1. Spring - formula 2. Lifting – formula 3. Pumping – formula not applicable ????(????) ≈ ????(???? ▯ ▯ ???? = ???? ▯ ▯▯▯ ▯ = lim ???? ???? Δ???? ▯→▯ ▯ ▯ ▯▯▯ ▯ = ????(????)???????? ▯ * How to Approach Spring Problems 1. Introduce x : Set x=0 at natural length. 2. Work done by F(x) : In stretching a spring b units beyond its natural length F(x) = -kx Hooke’s Law Ji-woo Kim 14072463 Math 113 Jan 25, 2016 * How to Approach Lifting Problems Set x = 0 at the bottom. x = amount of rope that has been pulled up ▯ ???? = ???? ???? ???????? ▯ * How to Approach Pumping Problems : Slice, Approximate, Sum and Limit 1. Slice 2. ???? :▯Work done to lift k-th layer out of the tank ???? ▯ ???? ∗ ▯ ▯ Chapter 7.2: Integration by Parts ???????????? = ???????? − ???? ???????? Order : LIATE (Logarithm, Inverse function, Algebraic, Trigonometric, Exponential) ???? = a function whose category comes first ???????? = the rest Ex) ln???? ???????? = ???? ???????????? − ???? ∗ 1 ???????? ???? = ???? ???????????? − ???? + ???? ???? = ln???? ???????? = 1 ???????? ▯ ???????? = ???????? ???? = ???? ▯ For Definite Integrals ▯ ▯ ???? ???????? = [????????] −▯ ???? ???????? ▯ ▯ ▯ Chapter 7.5: Integrating Rational Functions by Partial Fractions ????(????) = ???????????????????????????????? ???????????????????????????????? ????(????) ▯ * ▯???????? = ln ???? + ???? Ji-woo Kim 14072463 Math 113 Jan 25, 2016 * ▯ ???????? = ln ???? + ???? + ???? ▯▯▯ * ▯ ???????? = ln ???????? + ???? + ???? ▯▯▯▯ ▯ * ▯ ???????? = arctan???? + ???? ▯ ▯▯ * ▯???????? = arctan( ) + ???? ▯ ▯ ▯ Partial Fraction Decomposition ????(????) ????(????) ∶ only works for proper rational function ▯(▯) ▯(▯)∶ Proper if degree of A(x) < degree of B(x) : Improper if degree of A(x) > degree of B(x) Proper Rational Function 1. Factor D(x) completely into a product of linear factors and irreducible quadratic factors 2. Write an appropriate decomposition form ????(????) ▯ ▯ ▯ ???????? + ???? (???????? + ???????? + ????) ????▯ ????▯ ???? ▯ ???? ▯ + ???? ▯ = (???????? + ????) + (???????? + ????) ▯ + ⋯+ ???????? + ???? ▯ + ???????? + ???????? + ???? ???? ▯ + ???? ▯ ????▯???? + ???? ▯ + ???????? + ???????? + ???? ▯+ ⋯+ (???????? + ???????? + ????) ▯ 3. Determine unknown constant : Multiply each side of (*) by D(x) : Find unknowns 1) Equate corresponding coefficients A x ▯ B = ▯ x + B▯ ▯ 2) Assign a convenient value for x Improper Rational Function ????(????) ???? ???? = = polynomial + proper rational function (???????????????? ????????????????????????????????) ????(????) ▯▯▯▯▯▯▯▯▯ (▯) = Quotient + ▯(▯) Ji-woo Kim 14072463 Math 113 Jan 25, 2016 Ex) 3???? + 3???? − 5???? + ???? − 1 1 ???????? = 3???? + 1???????? + ???????? ???? + ???? − 2 ???? + ???? − 2 ▯ 1 = 3???? + 1???????? + ???????? (???? + 2)(???? − 1) 1 1 ▯ 3 3 = ???? + ???? + ???? + 2 ???????? + ???? − 1 ???????? 1 1 = ???? + ???? − ln ???? + 2 + ln ???? − 1 3 3 Chapter 7.8: Improper Integrals Definition: ▯???? ???? ???????? is improper if ???? = −∞ or b = ∞ OR if f(x) has an infinite ▯ discontinuity on [a,b] ▯ ▯ ???? ???? ???????? = l▯→▯ ???? ???? ???????? ▯ ▯ If the limit exists, the improper integral converges. If the limit doesn’t exist, the improper integral diverges. ▯ ▯ ???? ???? ???????? = l▯→▯ ???? ???? ???????? ▯▯ ▯ ▯ ▯▯ ▯ ???? ???? ???????? = l▯→▯ ???? ???? ???????? + ???? ???? ???????? ▯▯ ▯▯ ▯ Ex) ▯▯ ▯ 1 ???????? = lim 1???????? = lim ln ???? ▯= lim ln ???? − ln 1 = lim ln ???? = ∞ ???? ▯→▯ ???? ▯→▯ ▯ ▯→▯ ▯→▯ ▯ ▯ Chapter 8.2: Separation of Variables ???????? = ????(????) ???????? ???????? ???? ???? = ????(????) = ???? ???? ℎ ???? → separable differential equation ???????? Ji-woo Kim 14072463 Math 113 Jan 25, 2016 ???????? = ???? ???? ???????? ℎ ???? ???????? ℎ ???? = ???? ???? ???????? ???? ???? = ???? ???? + ???? Ex) ???????? ▯ ???????? = 2(1 + ???? )???? 1 ▯???????? = 2???? ???????? 1 + ???? 1 1 + ???? ▯???????? = 2???? ???????? arctan???? = ???? + ???? ▯ ???? = tan ???? + ???? Chapter 7.3: Integrating Trigonometric Functions sin???? ???????? = −cos???? + ???? cos???? ???????? = sin???? + ???? tan???? ???????? = −ln cos???? + ???? sec???? ???????? = ln sec???? + tan???? + ???? csc???? ???????? = ln csc???? − cot???? + ???? ▯ *sin???? ∗cos???? = sin ???? + ???? +sin ???? − ???? ▯▯ *sin???? ∗sin???? = ▯ cos ???? − ???? −cos ???? + ???? ▯ *cos???? ∗sin???? = ▯ cos ???? − ???? +cos ???? + ???? 1 sin????????cos???????? ???????? = sin ???????? − ???????? +sin ???????? + ???????? ???????? 2 Ji-woo Kim 14072463 Math 113 Jan 25, 2016 ???????????? ???? ???????????? ???? ????????, Case 1. m is odd, n is any -> Factor out ???????????? ???? and use ???????????? ???? = 1 − ???????????? ???? ▯ -> To write the integrand in terms of ???????????? ????, use ???? = cos???? for substitution method Case 2. m is any, n is odd ->Factor out ???????????? ????. ???????????????? ???????????? ???? ???????????? ▯▯▯ ???? ???????? -> use ???????????? ???? = 1 − ???????????? ???? and use ???? = sin???? for further substitution Case 3. m is even, n is even ▯ ▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯ -> use half-angle (double) identities (???????????? ???? = ▯ ,???????????? ???? = ▯ ) ▯ ▯ ???????????? ???? ???????????? ???? ????????, Case 1. m is odd, n is any ▯ ▯ -> Factor out ???????????? ???? ???????????? ???? and use ???????????? ???? = ???????????? ???? − 1 -> To write the integrand, use ???? = sec???? for substitution method Case 2. m is any, n is even ->Factor out ???????????? ???? and use ???????????? ???? = ???????????? ???? + 1 -> use ???? = tan???? for further substitution Case 3. m is even, n is odd -> ???????????? ???? = ???????????? ???? − 1 -> To write the integrand in terms of sec x, ▯ 1 ▯▯▯ ???? + 2 ▯▯▯ ???????????? ???????????? = ???? − 1 ???????????? ???????????????????? + ???? − 1 ???????????? ???? ???????? (Reduction Formula)

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