Winter 2008 - Enright's Class - Exam 1

Winter 2008- - 20E Midterm Solutions March 7 2008 1 Using methods of the course compute the surface area of the sphere of radius 5 above the 45

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Unformatted text preview: 20E Midterm Solutions March 7, 2008 1) Using methods of the course compute the surface area of the sphere of radius 5 above the 45 degree latitude line. Answer We paramaterize the area: x = 5 cos( θ ) sin( φ ) y = 5 sin( θ ) sin( φ ) z = 5 cos( φ ) . with ≤ φ ≤ π/ 4 and ≤ θ ≤ 2 π . For this parameterization in spherical coordinates: || T φ × T θ || = 25 sin( φ ) . Thus: A = Z Z S || T φ × T θ || dθdφ = Z π/ 4 φ =0 Z 2 π θ =0 25 sin( φ ) dθdφ = 50 π Z π/ 4 φ =0 sin( φ ) dφ = 50 π (1- √ 22) 2) Compute the volume of the intersection of the vertical unit cylinder and the sphere of radius 2 and center at (0 , , 0) . Answer We present a solution using cylindrical coordinates. Recall the Jacobian for the change of variables to cylindrical coordinates is r . We have: V = Z 1 r =0 Z 2 π θ =0 Z √ 1- r 2 z =- √ 4- r 2 rdzdθdr = 2 π Z 1 r =0 2 r p 4- r 2 dr = 2 π Z 4 u =3 √ udu = 4 3 π (4 3 / 2- 3 3 / 2 ) 1 3) Let F be the velocity vector field F...
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This note was uploaded on 04/30/2008 for the course MATH 20E taught by Professor Enright during the Winter '07 term at UCSD.

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Winter 2008- - 20E Midterm Solutions March 7 2008 1 Using methods of the course compute the surface area of the sphere of radius 5 above the 45

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