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Chapter 8 1. 1 Given the region bounded by the graphs of y = ln x , y = 0 , and x = e 2 . a. Use integration by parts to determine the area of the region. b. Find the volume of the solid generated by rotating the region about the line x = e 2 . c. Find the volume of the solid generated by rotating the region about the horizontal line y = −2 . Show your work, using integration by parts. Chapter 8 2 Given the region bounded by the graphs of y = ln x , y = 0 , and x = e 2 . 1. a. Use integration by parts to determine the area of the region. +2: integral e2 Area = ∫ ln x dx = x ln x − ∫ 1dx , using integration by +1: answer 1 parts with u = ln x & dv = dx . Area = (x ln x − x ) 1 = e 2 + 1 e2 b. Find the volume of the solid generated by rotating the region about the line x = e 2 . Volume = 2 +1: limits, constant, answer 2 π ∫ (e 2 − e y ) dy = π ∫ (e 4 − 2e 2 e y + e 2 y )dy = 2 0 +2: integral 0 2 1 1⎞ ⎛ ⎞ ⎛1 π ⎜ e 4 y − 2e 2 e y + e 2 y ⎟ = π ⎜ e 4 + 2e 2 − ⎟ = 41.577π 2 2⎠ ⎝ ⎠0 ⎝2 c. Find the volume of the solid generated by rotating the region about the horizontal line y = −2 . e 2 [ ] π ∫ (ln x + 2)2 − (2) 2 dx = 1 e2 [ +1: limits, constant, answer ] Volume = π ∫ (ln x ) + 4 ln x dx = 1 [ 2 π x(ln x )2 + 2 x ln x − 2 x π (6e 2 + 2) ] e2 1 +2: integrand = Chapter 8 3 3 2. Given the region bounded by the graphs of y = e − x , x = 0 , y = 0 , and x = b (b > 0) . a. Find the area of the region if b = 1 . b. Find the volume of the solid generated if the region is rotated about the xaxis, and b = 1 . c. Find the volume of the solid generated when the region is rotated about the x-axis and b = 10 . Chapter 8 4 3 2. Given the region bounded by the graphs of y = e − x , x = 0 , y = 0 , and x = b a. Find the area of the region if b = 1 . 1 Area = ∫ e − x dx = .8075 3 (b > 0) . +2: integral +1: answer 0 b. Find the volume of the solid generated if the region is rotated about the x-axis and b = 1 . +1: limits, constant, answer Volume = 1 ( ) π ∫ e− x 0 3 2 +2: integrand 1 dx = π ∫ e −2 x dx = 0.6907π ≈ 2.170 3 0 c. Find the volume of the solid generated when the region is rotated about the x-axis and b = 10 . 10 ( ) dx = π ∫ e Volume = π ∫ e − x 0 3 2 10 0 −2 x3 dx = .7088π ≈ 2.227 +2: integrand +1: limits, constant, answer Chapter 8 3. 5 Consider the graph of y = xe −x 3 . a. Set up the integral used to find the area under the graph of this curve between x = 0 and x = 2 . b. Identify u and dv for finding the integral using integration by parts. c. Evaluate the integral using integration by parts. Chapter 8 3. 6 Consider the graph of y = xe −x 3 . a. Set up the integral used to find the area under the graph of this curve between x = 0 and x = 2 . 2 Area = ∫ xe −x 3 +1: integrand +1: limits dx 0 b. Identify u and dv for finding the integral using integration by parts. −x c. u=x and dv = e 3 dx Evaluate the integral using integration by parts. Let u=x dv = e −x du = dx v = −3e 2 Then, ∫ xe −x 3 3 −x dx = −3 xe +1: dv = e −x 3 dx +2: uv − ∫ vdu dx 3 −x +1: u = x +2: antiderivatives 3 0 2 − ∫ − 3e −x 3 dx = ⎡− 3 xe − x 3 − 9e − x 3 ⎤ = −15e − 2 3 + 9 ≈ 1.299 ⎢⎣ ⎥⎦ 0 +1: answer Chapter 8 4. 7 Consider the graph of y = ln 2 x . a. Set up the integral used to find the area under the graph of this curve between x = 1 and x = 3 . b. Identify u and dv for finding the integral using integration by parts. c. Evaluate the integral using integration by parts. Chapter 8 8 Consider the graph of y = ln 2 x . 4. a. Set up the integral used to find the area under the graph of this curve between x = 1 and x = 3 . +1: integrand +1: limits 3 Area = ∫ ln 2 x dx 1 b. Identify u and dv for finding the integral using integration by parts. c. +1: u = ln 2 x u = ln 2 x and dv = dx Evaluate the integral using integration by parts. +1: dv = dx dv = dx u = ln 2 x Let 1 du = 2dx v = x 2x +2: uv − ∫ vdu Then, +2: antiderivatives +1: answer 3 1 ∫ ln 2 xdx = x ln 2 x − ∫ x ⋅ x dx 1 3 1 = [x ln 2 x − x ] 1 = 3 ln 6 − ln 2 − 2 ≈ 2.682 3 Chapter 8 9 Consider the region bounded by the x-axis, the line x = 5 , and the curve x2 . f ( x) = 3 2 2 x +9 5. ( ) a. Choose an appropriate trigonometric substitution to find the area of this region. Explain why you made this choice. b. Find the area of this region using the method of trigonometric substitution. c. Evaluate lim f ( x) . x →∞ Chapter 8 10 Consider the region bounded by the x-axis, the line x = 5 , and the curve x2 . f ( x) = 3 2 2 x +9 5. ( ) a. Choose an appropriate trigonometric substitution to find the area of this region. Explain why you made this choice. +1: chooses x = 3 tan θ Let x = 3 tan θ . Using this substitution will result in a denominator of the form (9(tan 2 θ + 1)) 2 = (9 sec 2 θ ) 3 3 +1: explanation 2 b. Find the area of this region using the method of trigonometric substitution. Note: x = 3 tan θ +2: Substitutes for x and dx +2: antiderivatives dx = 3 sec θ dθ Also: 3 tan θ = 0 ⇒ θ = 0 3 tan θ = 5 ⇒ θ = 1.03038 2 5 ∫ 0 1.03038 x2 (x 2 1.03038 ∫ 0 + 9) 3 2 dx = ∫ 0 tan 2 θ dθ = sec θ 1.03038 ∫ 0 ( 9 tan 2 θ 9 tan θ + 9 2 +1: answer ) sec 2 θ − 1 dθ = sec θ ( ln sec θ + tan θ − sin θ ) c. Evaluate lim f ( x) . 1.03038 0 3 ⋅ 3sec 2 θ dθ = 1.03038 ∫ ( sec θ − cos θ ) dθ = 0 ≈ 0.426 x →∞ Using L’Hopital’s rule, 1 x2 = lim =0 lim 1 3 2 x →∞ (x 2 + 9 ) 2 x →∞ 3 (x + 9 ) 2 2 +1: correct use of L’Hopital’s Rule +1: answer Chapter 8 11 Consider the region bounded by the x-axis, the line x = 2 , and the curve x3 . f ( x) = 4 + x2 6. a. Choose an appropriate trigonometric substitution to find the area of this region. Explain why you made this choice. b. Find the area of this region using the method of trigonometric substitution. c. Evaluate lim f ( x) . x →∞ Chapter 8 12 Consider the region bounded by the x-axis, the line x = 2 , and the curve x3 . f ( x) = 4 + x2 6. a. Choose an appropriate trigonometric substitution to find the area of this region. Explain why you made this choice. +1: Chooses x = 2 tan θ Let x = 2 tan θ . Using this substitution will result in a +1: explanation denominator of the form 4(tan θ + 1) = 4 sec θ b. Find the area of this region using the method of trigonometric substitution. 2 2 ∫ dx = 4 + x2 0 π π x3 4 ∫ 0 8 tan 3 θ 4 + 4 tan 2 θ π 4 2 +2: Substitutes for x and dx +2: antiderivative ⋅ 2 sec θ dθ = 4 ( 2 +1: answer ) 3 2 ∫ 8 tan θ secθ dθ = 8 ∫ sec θ − 1 secθ tan θ dθ = 0 0 ⎡2 2 2⎤ 8 ∫ u 2 − 1 du = 8⎢ − 2+ ⎥ 3⎦ 1 ⎣ 3 c. Evaluate lim f ( x) 2 ( ) x →∞ Using L’Hopital’s Rule, x3 3x = lim = lim −1 x →∞ x →∞ (4 + x 2 ) 2 4 + x2 lim 3x(4 + x ) 2 1 2 =∞ x →∞ Therefore, this limit does not exist. +1: Uses L’Hopital’s Rule +1: answer Chapter 8 7. 13 Consider the function h( x) = a. x − cos x . x Can you use L’Hopital’s Rule to find lim h( x) ? Explain your reasoning. x →∞ b. Find lim h( x) analytically. x →∞ c. Set up the integral needed to find the area under the graph of h(x) between x = 1 and x = π (above the x-axis). Chapter 8 7. a. Consider the function h( x) = 14 x − cos x . x Can you use L’Hopital’s Rule to find Explain your reasoning. lim h( x) ? x →∞ ∞ Yes; lim h( x) is of the form . Even though cosine ∞ x →∞ oscillates, the x term dominates and the numerator goes to infinity. b. Find lim h( x) analytically. +1: yes +2: explanation x →∞ x − cos x ⎛ x cos x ⎞ = lim ⎜ − ⎟= lim x x ⎠ x →∞ x →∞ ⎝ x cos x 1 − lim = 1− 0 = 1 lim x x →∞ x →∞ c. Set up the integral needed to find the area under the graph of h(x) between x = 1 and x = π (above the x-axis). π x − cos x dx Area = ∫ x 1 +2: separation into fractions +1: answer +2: integrand +1: limits Chapter 8 8. 15 Consider the function g ( x) = a. ex −1 . x2 Can you use L’Hopital’s Rule to find lim g ( x) ? Explain your reasoning. x →0 + b. Find lim g ( x) analytically. x→0 + c. Describe the type of indeterminate form involved in directly evaluating 2 −x lim x e . Evaluate this limit using L’Hopital’s Rule. x →∞ Chapter 8 8. a. 16 Consider the function g ( x) = ex −1 . x2 Can you use L’Hopital’s Rule to find lim g ( x) ? x→0 + Explain your reasoning. lim g ( x) is of the form Find lim g ( x) analytically. Yes; x→0 + b. 0 . 0 +1: yes +2: explanation x→0 + ex −1 ex 1 = = . This limit does not exist. lim lim 0 x2 x→0 + x →0 + 2 x c. Describe the type of indeterminate form involved in directly evaluating lim x 2 e − x . Evaluate this limit x →∞ using L’Hopital’s Rule. The indeterminate form is the form ∞ ⋅ 0 . x2 2x 2 2 −x = = lim x = lim x = 0 x e lim lim x x →∞ x →∞ e x →∞ e x →∞ e +2: Uses L’Hopital’s rule correctly +1: answer +1: indeterminate form +2: answer Chapter 8 9. 17 2 . x3 Consider the function f ( x) = 1 a. The evaluation of 2 ∫x 3 dx = 0 is incorrect. Explain why. −1 b. Determine whether this improper integral converges or diverges. Show your work to justify your conclusion. c. Evaluate ∞ ∫ f ( x)dx 1 analytically. Chapter 8 18 Consider the function f ( x) = 9. 1 a. 2 ∫x The evaluation of 3 2 . x3 dx = 0 is incorrect. Explain +2: explanation −1 why. 2 has an infinite discontinuity at x = 0 . x3 Determine whether this improper integral converges or diverges. Show your work to justify your conclusion. b. +1: uses limits 1 2 ∫x −1 3 t 1 2 2 dx + lim ∫ 3 dx = 3 s →0 + s x −1 x dx = lim ∫ t →0 − 1 t −1 −1 + lim 2 lim 2 t → 0 − x −1 s →0 + x s Therefore, this improper integral diverges. +1: antiderivative +1: diverges ∞ c. Evaluate ∫ f ( x)dx analytically. 1 ∞ ∫ 1 t t 2 −1 f ( x)dx = lim ∫ 3 dx = lim 2 = 1 t →∞ 1 x t →∞ x 1 +2: uses limit +1: antiderivative +1: answer Chapter 8 10. 19 Consider the function g ( x) = 1 . ( x − 2 )2 3 a. Explain why ∫ g ( x)dx is an improper integral. 0 3 b. Determine whether ∫ g ( x)dx 0 justify your conclusion. ∞ c. Evaluate ∫ g ( x)dx 4 . converges or diverges. Show your work to Chapter 8 10. 20 Consider the function g ( x) = 1 . ( x − 2 )2 3 a. Explain why ∫ g ( x)dx is an improper integral. +2: explanation 0 g (x) has an infinite discontinuity at x = 2 . 3 b. Determine whether ∫ g ( x)dx converges or diverges. 0 Show your work to justify your conclusion. 3 t ∫ g ( x)dx = lim ∫ (x − 2) t →2− 0 0 3 1 2 dx + lim ∫ s →2+ s t 1 (x − 2) 2 dx = 3 −1 −1 =∞ + lim lim t →2− x − 2 0 s→2+ x − 2 s +1: uses limits +1: antiderivative +1: answer It diverges. ∞ c. Evaluate ∫ g ( x)dx +2: integrand 4 t 1 ∫ lim (x − 2 ) t→∞ 0+ 2 dx = lim 1 1 = . 2 2 t→∞ −1 = x−2 4 +1: limits +1: answer