Explanation Formula used: Let C be the curve with the parametric equation x = f ( t ) , y = g ( t ) , for a ≤ t ≤ b , where f ′ and g ′ is continuous on [ a , b ] and C does not intersect itself, except possibly at its endpoints. Then, the area of the surface obtained by revolving C about the x -axis is, S = ∫ a b 2 π g ( t ) f ′ ( t ) 2 + g ′ ( t ) 2 d t The area of the surface obtained by revolving C about the y -axis is, S = ∫ a b 2 π f ( t ) f ′ ( t ) 2 + g ′ ( t ) 2 d t Calculation: The given parametric equation is x = cos 3 t , y = sin 3 t , for 0 ≤ t ≤ π 2 . Here, x = f ( t ) = cos 3 t and y = g ( t ) = sin 3 t . Differentiate f ( t ) and g ( t ) with respect to t as follows. f ′ ( t ) = d d t ( cos 3 t ) = − 3 cos 2 t sin t g ′ ( t ) = d d t ( sin 3 t ) = 3 sin 2 t cos t Substitute f ′ ( t ) = − 3 cos 2 t sin t and g ′ ( t ) = 3 sin 2 t cos t in S = ∫ a b 2 π g ( t ) f ′ ( t ) 2 + g ′ ( t ) 2 d t

Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134763644

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 12.1, Problem 108E

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The surface area of the curve x=cos3t,y=sin3t, for 0≤t≤π2, about the x-axis.

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Chapter 12.1, Problem 107E

Chapter 12.1, Problem 109E

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u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (ū+v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅w) Support your answer mathematically or a with a written explanation. d) If possible, find u. (vxw) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

Question 3 (6 points) u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (u + v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅ w) Support your answer mathematically or a with a written explanation. d) If possible, find u (v × w) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

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Chapter 12 Solutions

Calculus: Early Transcendentals (3rd Edition)

Ch. 12.1 - Identify the graph generated by the parametric...Ch. 12.1 - Prob. 2QC Ch. 12.1 - Describe the curve generated by x = 3 + 2t, y = 12...Ch. 12.1 - Find parametric equations for the line segment...Ch. 12.1 - Use Theorem 12.1 to find the slope of the line x =...Ch. 12.1 - Use the arc length formula to find the length of...Ch. 12.1 - Explain how a pair of parametric equations...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Give parametric equations that generate the line...

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The surface area of the curve x = cos 3 t , y = sin 3 t , for 0 ≤ t ≤ π 2 , about the x -axis.

Solution Summary: The author calculates the surface area of the curve with the parametric equation x=mathrmcos3t, y=