QUESTION
MATHS 2026 ASSIGNMENT 2
Submission date : 12.00 noon on Monday 4 June 2012 in the designated Assignment
Box in the Maths Department foyer.
Cheating or failure to identify your collaborators on the assignment cover sheet will
result in a zero mark. Make a copy of your assignment before submission.
Late submission without prior approval will result in the loss of 10% of total marks per
day late.
Question 1. Find the volume under the surface
z = 4 ! x
2
! y
and above the
xy
plane
bounded on the sides by the surfaces
y = ±Cosx !
2
!
2
∀ x ∀
(10 marks)
Question 2. (a) Evaluate
!
∀
0
!
∀
0
dxdy
Cosh(x
2
(b) Find the volume between the ellipsoid
4x
+ y
2
)
2
+ 4y
2
+ z
= 80
and the
paraboloid
z = 2x
2
+ 2y
(10 marks)
2
Question 3. Using spherical coordinates, calculate the moment of inertia about the
z
axis of the uniform body that lies between the sphere
x
and the cone
z = x
!
∀
2
+ y
2
#
∃
1
2
.
(10 marks)
2
Question 4. (a) Derive an explicit expression for the surface area of an axially
symmetric surface
r = r(!)
as an integration in spherical coordinates.
(b) Hence find the surface area of that piece of the sphere
x
2
+ y
!
2
2
which lies between the planes
z = b
and
z = c
where
0 ! b < c ! a
.
(10 marks)
(Question 5 over page)
2
+ z
+ y
2
2
= a
+ z
2
2
= a
2
Question 5. Consider the region
R
bounded by the circles
x
2
+ y
2
= Ax
,
x
= Bx
,
x
= Dy
where B > A , D > C . Use the change of
variables
2
+ y
2
= Cy
and
x
u =
2
x
x
2
+ y
+ y
2
2
v =
y
to evaluate the integral
!
x
dxdy
(x
2
+ y
2
2
3
R
+ y
)
2
(10 marks)
2
+ y
2
SOLUTION
Answer1.
We have
Integrating in limits,
V=2(8-8/3)=32/3=10.67
Answer2.
(a)
First integrating with respect to x.
Then du=2xdx ,
LE84
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