Question 1
Q1a. Find the positive constant A given that the volume of the solid bounded by the paraboloid,
2 2 x y z A ? ? ?
in the first octant is
32? . (8 marks)
Q1b. Use Green’s Theorem to evaluate
sin5 5
C
ydx xdy ? ?
around the boundary curve
C
of the
region R, where R is the triangle formed by the point (0, 0), (1, 1) and (1, 3). (7 marks)
Question 2
Q2a. For the vector field
F i j k ? x y z y xy x xz , , 2 2 . ? ? ? ? ? ? ?
(i) Show that
? ?
1
.
2
? ? ?? F = F div (5 marks)
(ii) Determine whether if the vector field
F
is conservative or not. (3 marks)
Q2b. Find the work done by the force field
F i j ? x y y x , 4 2 ? ? ?
in moving a particle along a
circle
2 2
x y ? ?1
from (0, 1) to (1, 0).
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