# Mathematical problems for self assessment

**Sample Problems**

**APPLICATIONS OF INTEGRATION II**

**1.** Find the length of the curve between and

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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 .

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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.

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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.

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**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 = ].