M25008SelSol9

M25008SelSol9 - Mathematics 250 Fall 2008 Selected...

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Unformatted text preview: Mathematics 250 Fall 2008 Selected Solutions Assignment # 9 MVM, 8.3.2 : Calculate the following line integrals. a) R C xy 3 dx , where C is the unit circle, C = x y : x 2 + y 2 = 1 , oriented counterclockwise. b) R C zdx + xdy + ydz , where C is the line segment from 1 2 to 1- 1 3 c) R C y 2 dx + zdy- 3 xydz , where C is the line segment from 1 1 to 2 3- 1 . d) R C ydx , where C is the intersection of the unit sphere with the plane x + y + z = 0, oriented counterclockwise as viewed from above. Response: a) We can parametrize C by γ ( t ) = cos t sin t , for 0 ≤ θ ≤ 2 π . This parametrization is consistent with the specified orientation. Then γ * ( xy 3 dx ) = γ * ( x ) γ * ( y ) 3 γ * ( dx ) = (cos t )(sin 3 t )(- sin tdt ) =- sin 4 t cos tdt. Thus, we have Z C xy 3 dx = Z 2 π- sin 4 t cos tdt = (- sin 5 t 5 | 2 π = 0- 0 = 0 . Indeed, this result could have been predicted by symmetry: the curve C is symmetric under reflection in the y-axis, and the contribution of an angular interval in the right half plane is cancelled by the contribution of its reflection in the y-axis. b) We can parametrize C by γ ( t ) = 1 2 + t 1- 1 3 - 1 2 = t 1- 2 t 2 + t , for 0 ≤ t ≤ 1. Then Z C ydx = Z 1 (1- 2 t ) dt = ( t- t 2 ) | 1 = 1- 1- (0- 0) = 0 . c) We can parametrize C by γ ( t ) = 1 1 + t 2 3- 1 - 1 1 = 1 + t 3 t 1- 2 t , for 0 ≤ t ≤ 1. Then we have Z C y 2 dx + zdy- 3 xydz = Z 1 ( (3 t ) 2 dt + (1- 2 t )(3 dt )- 3(1 + t )(3 t )(- 2 dt ) ) = Z 1 (27 t 2 + 12 t + 3) dt = (9 t 3 + 6 t 2 + 3 t ) | 1 = (9 + 6 + 3)- (0 + 0 + 0) = 18 . d) To parametrize this curve, we follow the suggestion in the book, and construct an orthonormal basis for the plane in question. Let P be the plane defined by x + y + z = 0. One vector in P is the vector 1 f 1 = 1 √ 2 1- 1 , which spans the intersection of P with the ( x, y )-plane. We want a second vector in P orthogonal to f 1 . A little thought will produce the vector 1 √ 6 1 1- 22 ....
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M25008SelSol9 - Mathematics 250 Fall 2008 Selected...

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