Name —Surname
Std.No:
1
6
5
4
3
2
Duration : 120 msnutes
21
0-1) Reverse the order of the integrationand evaluate
Jsin(x2y1xdy
2
Osy<2
2
21
21
f fsin(x 2gxdy = f
sin(xl klydx =
dx =
2
=—cosl+l=l—cosl
03M.2017
T(TAL
0-2) Use triple integral to find the volume of the solid region which lies between parabola
: = x2
and the cylinder x' 4 y'
(z
0).
C)= OSrS2
rdzdrdO =
VOI
r 3drdO =
000
4(2r) = 8m
0-3) (10 points)Evaluatethe line integral[fle
oriented boundary of the following region.
By Green's Formula
+ xJ + ey2
3(x2 +
) dA = J 3r JdrdO
01
306-1)
4
458
x —
+ (xJ + e/
where C is
positively;
2)' + z) dS where S is the firstoctant
04) (20points)Evaluate
z 2
portion of the piano x 4
2)dS•
+ Id,d
20
4 4x4 x'
+ 12 4x)dr
dx =
x+y+zs:
VS 40 20
—
412x—2x2
e
y-2-x
Q-5) Given
3xzi + 4xyj+zxk
. Then evaluate
a) (6 points)divF V •F
324 4x4 x 3z+5x
divF
b) (8 points) curl F = V x F
curl F
öx O'
= j (O) j (2 3x) + i (4y) = (3x—2) j +
(3:
3.8? 4xy 2x
C) (6 points)
V •(V F)
F)
(3r
Z) + —(4y)
O
Q-6) (20 points)Use Divergence Theorem to evaluate the flux
= yxi+y
field F(x,y,z)
j+zk
F •n
of the vector
accross the surface of the region bounded by planes
x = 0, x = 2,y = 0, y = 4, z =0 and y+z= 6 (theregiongiven below).
z
6
eY+Z=6
x
VFdV
Divergence theorem
f F.;adS
JJ
000
jj(6
—y) (3y + 1) dydx
00
(3y+1)dzdydx
J (17y + 6 —3y 2) dydx
00
4
2
17
= f —y3 +—y 2 +6y
2
2
J 96dx =96x1 0 —192