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# Mathematical problems for self assessment

###### 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 = ].