Mathematical problems for self assessment
Sample Problems
APPLICATIONS OF INTEGRATION II
1. Find the length of the curve 
between 
and 
[Ans. 
]
2. Find the length of the cycloid parameterised by 
between 
and 
.
[Ans. 8]
3. Find the area of the surface of revolution generated by the curve 
when it is rotated about the
x–axis between 
and 
.
[Ans.
.
4. Find the area of the surface of revolution generated by the curve 
when it is rotated about the x-axis between 
and 
.
[Ans.
]
5. Find the moment of inertia of a thin rod of length 
and mass M about an axis normal to its length and acting through the rod at a distance 
from one end.
[Ans.
]
6. Find the moment of inertia of an annular lamina of mass M with inner radius 
and outer radius 
about a line L through its centre which is normal to its plane.
[Ans.
]
7. Find the moment of inertia of a thin rectangular plate of mass M, length 
and width 
about an axis through its centre of gravity which is normal to its plane.
[Ans.
]
8. Find the moment of inertia of a thin square plate of mass M and side length about a line 
which lies in the same plane as the plate and which is parallel to, and at a distance 
from, a line L that passes vertically through the centre of gravity of the plate.
[Ans.
]
.
9. Find both the mean and root-mean-square values of the following functions over the given intervals:
(i) 
,
(ii) 
.
[Ans. (i) mean value = 19/3, r.m.s. value=
, (ii) mean value = 2+
, r.m.s. value = 
].