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# Chapter 5 study guide Math 1221

GWU

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## Popular in Calculus with PreCalculus II

## Popular in Mathematics (M)

This 9 page Study Guide was uploaded by Jamie Elliott on Tuesday May 3, 2016. The Study Guide belongs to Math 1221 at George Washington University taught by Roosevelt in Spring 2016. Since its upload, it has received 28 views. For similar materials see Calculus with PreCalculus II in Mathematics (M) at George Washington University.

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Date Created: 05/03/16

Math 1221: Calculus with Pre-Calculus II Study Guide Specified Objective: Chapter 5 5.1 through 5.3, and 5.5 Chapter 5: 5.1 In this section, we discussed finding the area between two curves. We used the drawings of graphs and several equations in order to find the area. ???? The Blue shaded area equals to the area under∫f???? ???? ???????? ???? ???? The Black straight lines equal to the are unde???? ???? ???? ???????? For f(x) and g(x) are continuous on [a, b], and f(x) ≥ g(x), then the area between y = f(x) and y = g(x) and bounded by x = a and x = b is the Ar∫a ???? ???? − ???? ???? ????????] ???? Categories: A) Bounds are given Ex: y = sin(x), y = x ; x = pi/2, x = pi Find the area bounded by the curves 1) Simple Sketch – This shows which one is on top. B) Bounds not given. 1) Set equations equal to each other and solve for x. Ex: y = (x-2) , y = x. Find the area between them. 2 (x-2) 2 x (x-2) – x = 0 (x - 4x + 4 – x) = 0 x – 5x + 4 = 0 (factor) (x – 4)(x – 1) = 0 x = 4, x = 1 2) Sketch. C) The “y” integrals or choice of your x. Ex: x = y , ???? = √2 − ???? 1) (If y=f(x)&y=g(x),∫ ???????? ; x=h(y), x=k(y)∫ ????????) Must have the same format, both “y =”or both “x =”. If not, then choose one equation and turn it into the other type of equation. For the example, use “x =”; take ????√=2 − ???? and solve for x. y = 2 – x (subtract 2) 2 2 y – 2 = x x = 2 – y 2) Set the two equations equal to each other. 4 2 y = 2 – y y + y – 2 = 0, Let u = y 2 2 u + u – 2 = 0, factor (u + 2)(u – 1) = 0 u = -2, u = 1; or y = -2 (not a solution), y = ±1; y = 1 is the only true/real solution. 3) Sketch. D) Multiple intersections 3 Ex: y = x , y = x x = x x - x = 0 x(x – 1) = 0 x = 0, x – 1 = 0 x = 1 x = ±1 Π Chapter 5: 5.2 In this section, we discussed Volume, pertaining to discs, the volume of a solid; and the method of Washer—Hollow Parts. In section 5.3, we will also be discussing Volume, pertaining to Cylindrical Shells. 5 2 ???? 2 2 Ex: ∫0 ???????? ????????, r = 2 ∫ ???????? ???????? ???????? ℎ 5 = ∫0 4???? ???????? = 4Π y|05 = 20Π Ex: Find the volume (V) of the solid generated by rotating the curve, y = x , 0 ≤ x ≤ 3, about the y-axis. (About x = 0) -If rotating about the y-axis, the thickness is dy. -If rotating about the x-axis, the thickness is dx. 9 9 = ∫ ???? ????idy (r is the opposite variable)∫=???????? ???????? (convert x to y, using given equation) 0 0 2 (y = x ) V= ∫ 9???? ???? ???????? = ???? ???? |0 → 9 = 81???? 0 2 2 1) (As mentioned before) -If rotated upon the y-axis, thickness is dy. -If rotated upon the x-axis, thickness is dx. 2) Put in limit∫???? ???????? (y limits) ∫r???????????? (x limits) ???? ???? ???? ???? 3) ∫ ???????? ???????? , or∫ ???????? ???????? ???? ???? ???? 4) Convert,∫ ???????? ????????, x to y using the given equation where x = f(y) ???? ???? 2 ∫ ???? (???? ???? ) ???????? ???? ???? 2 Convert,∫???? ???????? ????????, y to x using the given equation where y = g(x) ???? 2 ∫???? ???? (???? ???? ) ???????? The is what the Washers method Chapter 5: 5.3 Within this section, we discussed Volume, pertaining to Cylindrical Shells. Surface Area of a Cylindrical Shell: 2pi * r * h Ex: Find the volume generated by rotating the region bounded by y = x , y = 0, x = 1, and x = 2, about the y-axis remember the “h” variable. 2 2 5 ???? = 2???? ∫ ???? ∗ ???? ???????? = 2???? ∫ ???? ???????? = 2???? ( )|1 → 2 = 2???? [( ) − ] = 2???? ( ) = 31 1 1 5 5 5 5 Chapter 5: 5.5 In this section, we reviewed the Average Value of a Function. “What’s the average y- coordinate of the function, the average value of a continuous function?” ∑ ???? ????−???? ????−???? l????→∞ ????=1????(???? i, where n = to the number of rectangles; ∆???? = ???? , ???? = ∆???? ???? ???? ???????? ???? ???? ???????? ∆???? lim ∑ ????=1 ????−???? = lim ∑????=1 ????−???? ????→∞ (∆???? ) ????→∞ ???? F a.v.= Average Value = 1/b-a ∫???? ???? ???? ???????? Example #18 from 5.5 : The velocity, v, of blood that flows in a blood vessel with radius, R, and length, l, at a distance, r, from the center is V(r)= P/4ɱl (R – r ); P = Blood pressure, ɱ = Viscosity -Everything is fixed, except r 2 2 -V(r) = C (R – r ) a) Max velocity: V’ = -2Cr = 0 (where r = 0 is center), V” = -2C (local max) b) Average velocity: Vav 1/R-0 (where R is outer boundary, 0 is center∫ ???????? ???? − 0 2) ????( 2 2) 2 3 ???? ???????? = 1/R ∫0 ???????? − ???????? ???????? = 1/R [CR r – Cr /3]|0R = 1/R [CR – CR /3 – 0] = 1/R (2/3 CR ) = 2/3 CR 2 Average value = 2/3 V, max

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