Page 57 - Engineering Mathematics Workbook_Final

Vector Calculus

(c) -1 and 3 (d) 2 and 2 along the path : c x t , y t = 3 ,
=
2
2
2
12. Let C be the circle (x − ) 1 + y = 1, 0 t   1 is _____________
oriented counter clockwise. Then, the [JAM MA 2016]
value of the line integral

C  − 4 xy dx + 3 x dy is 16. Let C be the boundary of the region
4
2
3 enclosed by y x= , y = x + 2 and
x = 0. Then the value of the line
(a) 6 (b) 8
3
integral C  ( xy − y 2 ) dx − x dy ,
(c) 12 (d) 14 where C is traversed in the counter

[JAM MA 2019] clockwise direction, is ____________
ur [JAM MA 2016]
$
$
2
) 2y i +
F x
13. Let ( , , y z = $ x j + xyk
and let C be the curve of intersection of 17. Let T be the smallest positive real
the plane x + + = 1 and cylinder number such that the tangent to the helix
y
z
t
$
x + 2 y 2 1 = . Then the value of cost i + $ sint j + $ 2 k at t = T is
C  ur  r orthogonal to the tangent at t= =0. Then
F dr is
ur
$
the line integral of F = x i − $ y j along
3 the section of the helix from t = 0 to t =
(a)  (b) T is ________
2
[JAM MA 2017]
(c) 2 (d) 3 ur $ $
)
+
F
[JAM MA 2019] 18. Let ( , x y = − y i x j and let C be
14. Evaluate C  yzdx + zxdy + xydz the ellipse x 2 + y 2 = 1 oriented
16
9
=
where C is the are of curve x b cost , counter clockwise. Then the value of
at ur r

=
y b sint , z = from the point C  F dr (round off to 2 decimal
2
0
intersects z = to the point it intersects places) is ________
=
z a. [JAM MA 2019]
ur
$
$
15. Let F = x i + ( x + y 3 ) j be a vector 19. Let C be the boundary of region x 1 y 2 
)
R =
y 
: 1
−
−
1,0  
(  ,x y 
2
R
)
0
field for all ( ,x y with x and oriented in the counter-clockwise
r direction. Then the value of
$
r = x i + $ y j . Then the value of the line  ydx + 2xdy is
r
ur
C   C
integral F dr from (0, 0) to (1, 1)

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