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ch14_05 - P1 OSO/OVY GTBL001-14-nal P2 OSO/OVY QC OSO/OVY...

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14-51 SECTION 14.5 . . Curl and Divergence 1165 16. C ( y sec 2 x 2) dx + (tan x 4 y 2 ) dy , where C is formed by x = 1 y 2 and x = 0 17. C x 2 dx + 2 x dy + ( z 2) dz , where C is the triangle from (0 , 0 , 2) to (2 , 0 , 2) to (2 , 2 , 2) to (0 , 0 , 2) 18. C 4 y dx + y 3 dy + z 4 dz , where C is x 2 + y 2 = 4 in the plane z = 0 19. C F · d r , where F = x 3 y 4 , e x 2 + z 2 , x 2 16 y 2 z 2 and C is x 2 + z 2 = 1 in the plane y = 0 20. C F · d r , where F = x 3 y 2 z , x 2 + z 2 , 4 xy z 4 and C is formed by z = 1 x 2 and z = 0 in the plane y = 2 In exercises 21–26, use a line integral to compute the area of the given region. 21. The ellipse 4 x 2 + y 2 = 16 22. The ellipse 4 x 2 + y 2 = 4 23. The region bounded by x 2 / 3 + y 2 / 3 = 1. (Hint: Let x = cos 3 t and y = sin 3 t ) 24. The region bounded by x 2 / 5 + y 2 / 5 = 1 25. The region bounded by y = x 2 and y = 4 26. The region bounded by y = x 2 and y = 2 x 27. Use Green’s Theorem to show that the center of mass of the region bounded by the positive curve C with constant den- sity is given by ¯ x = 1 2 A C x 2 dy and ¯ y = − 1 2 A C y 2 dx , where A is the area of the region. 28. Use the result of exercise 27 to find the center of mass of the region in exercise 26, assuming constant density. 29. Use the result of exercise 27 to find the center of mass of the region bounded by the curve traced out by t 3 t , 1 t 2 , for 1 t 1, assuming constant density. 30. Use the result of exercise 27 to find the center of mass of the region bounded by the curve traced out by t 2 t , t 3 t , for 0 t 1, assuming constant density. 31. Use Green’s Theorem to prove the change of variables formula R dA = S ( x , y ) ( u ,v ) du d v, where x = x ( u ,v ) and y = y ( u ,v ) are functions with continu- ous partial derivatives. 32. For F = 1 x 2 + y 2 y , x and C any circle of radius r > 0 not containing the origin, show that C F · d r = 0. In exercises 33–36, use the technique of example 4.5 to evaluate the line integral. 33. C F · d r , where F = x x 2 + y 2 , y x 2 + y 2 and C is any posi- tively oriented simple closed curve containing the origin 34. C F · d r , where F = y 2 x 2 ( x 2 + y 2 ) 2 , 2 xy ( x 2 + y 2 ) 2 and C is any positively oriented simple closed curve containing the origin 35. C F · d r , where F = x 3 x 4 + y 4 , y 3 x 4 + y 4 and C is any posi- tively oriented simple closed curve containing the origin 36. C F · d r , where F = y 2 x x 4 + y 4 , x 2 y x 4 + y 4 and C is any posi- tively oriented simple closed curve containing the origin 37. Where is F ( x , y ) = 2 x x 2 + y 2 , 2 y x 2 + y 2 defined? Show that M y = N x everywhere the partial derivatives are defined. If C is a simple closed curve enclosing the origin, does Green’s Theorem guarantee that C F · d r = 0? Explain. 38. For the vector field of exercise 37, show that C F · d r is the same for all closed curves enclosing the origin. 39. If F ( x , y ) = 2 x x 2 + y 2 , 2 y x 2 + y 2 and C is a simple closed curve in the fourth quadrant, does Green’s Theorem guarantee that C F · d r = 0? Explain.
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