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LINE INTEGRALS 4;2 x y dx y x dy along (a) the parabola y2 x, (b) a straight line, (c) straight lines 10.35. Evaluate
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1;1
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from 1; 1 to 1; 2 and then to 4; 2 , (d) the curve x 2t2 t 1; y t2 1. Ans: a 34=3; b 11; c 14; d 32=3 10.36. Evaluate 2x y 4 dx 5y 3x 6 dy around a triangle in the xy plane with vertices at 0; 0 ; 3; 0 , 3; 2 traversed in a counterclockwise direction. Ans. 12
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10.37. Evaluate the line integral in the preceding problem around a circle of radius 4 with center at 0; 0 . Ans: 64 10.38. (a) If F x2 y2 i 2xyj, evaluate F dr along the curve C in the xy plane given by y x2 x from the point 1; 0 to 2; 2 . Ans. (a) 124/15
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(b) Interpret physically the result obtained.
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LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS
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[CHAP. 10
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10.39. Evaluate
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2x y ds, where C is the curve in the xy plane given by x2 y2 25 and s is the arc length Ans: 15
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parameter, from the point 3; 4 to 4; 3 along the shortest path. 10.40. If F 3x 2y i y 2z j x2 k, evaluate
2 3 C
F dr from 0; 0; 0 to 1; 1; 1 , where C is a path consisting
of (a) the curve x t; y t ; z t , (b) a straight line joining these points, (c) the straight lines from 0; 0; 0 to 0; 1; 0 , then to 0; 1; 1 and then to 1; 1; 1 , (d) the curve x z2 ; z y2 . Ans: a 23=15; b 5=3; c 0; d 13=15 10.41. If T is the unit tangent vector to a curve C (plane or space curve) and F is a given force eld, prove that under appropriate conditions
F dr
F T ds where s is the arc length parameter.
Interpret the result
physically and geometrically. GREEN S THEOREM IN THE PLANE, INDEPENDENCE OF THE PATH 10.42. Verify Green s theorem in the plane for x2 xy3 dx y2 2xy dy where C is a square with vertices at 0; 0 ; 2; 0 ; 2; 2 ; 0; 2 and counterclockwise orientation. 10.43. Evaluate the line integrals of (a) Problem 10.36 and
Ans.
common value 8
(b) Problem 10.37 by Green s theorem.
10.44. (a) Let C be any simple closed curve bounding a region having area A. Prove that if a1 ; a2 ; a3 ; b1 ; b2 ; b3 are constants, a1 x a2 y a3 dx b1 x b2 y b3 dy b1 a2 A
(b) Under what conditions will the line integral around any path C be zero 10.45. Find the area bounded by the hypocycloid x2=3 y2=3 a2=3 . [Hint: Parametric equations are x a cos3 t; y a sin3 t; 0 @ t @ 2.] 10.46. If x  cos ; y  sin , prove that
Ans.
(b) a2 b1
Ans:
3a2 =8
x dy y dx 1 2 d and interpret. 2 x3 x2 y dx xy2 dy, where C is the boundary of the region
10.47. Verify Green s theorem in the plane for
enclosed by the circles x2 y2 4 and x2 y2 16. 2;1 10.48. (a) Prove that
1;0
Ans:
common value 120
2xy y4 3 dx x2 4xy3 dy is independent of the path joining 1; 0 and 2; 1 . Ans: b 5
(b) Evaluate the integral in (a). 10.49. Evaluate =2; 1 .
2xy3 y2 cos x dx 1 2y sin x 3x2 y2 dy along the parabola 2x y2 from 0; 0 to Ans. 2 =4
10.50. Evaluate the line integral in the preceding problem around a parallelogram with vertices at 0; 0 ; 3; 0 , 5; 2 ; 2; 2 . Ans: 0 10.51. (a) Prove that G 2x2 xy 2y2 dx 3x2 2xy dy is not an exact di erential. (b) Prove that e y=x G=x is an exact di erential of  and nd . (c) Find a solution of the di erential equation 2x2 xy 2y2 dx 3x2 2xy dy 0. Ans: b  e y=x x2 2xy c; c x2 2xy ce y=x 0
CHAP. 10]
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