CALCULATITON IN MATHS

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   ,

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