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Mathematical problems for self assessment

Sample Problems
APPLICATIONS OF INTEGRATION II

 

 

 

1. Find the length of the curve image placeholder between image placeholderand image placeholder

[Ans. image placeholder]

 

2. Find the length of the cycloid parameterised by image placeholder between image placeholder and image placeholder.image placeholder

[Ans. 8]

 

3. Find the area of the surface of revolution generated by the curve image placeholder when it is rotated about the

x–axis between image placeholderand image placeholder.

[Ans.image placeholder.

 

4. Find the area of the surface of revolution generated by the curve image placeholderwhen it is rotated about the x-axis between image placeholder and image placeholder.

[Ans.image placeholder]

 

5. Find the moment of inertia of a thin rod of length image placeholderand mass M about an axis normal to its length and acting through the rod at a distance image placeholderfrom one end.

[Ans.image placeholder]

 

6. Find the moment of inertia of an annular lamina of mass M with inner radius image placeholderand outer radius image placeholder about a line L through its centre which is normal to its plane.

[Ans.image placeholder]

 

7. Find the moment of inertia of a thin rectangular plate of mass M, length image placeholderand width image placeholderabout an axis through its centre of gravity which is normal to its plane.

[Ans.image placeholder]

 

8. Find the moment of inertia of a thin square plate of mass M and side length image placeholderabout a line image placeholder which lies in the same plane as the plate and which is parallel to, and at a distance image placeholderfrom, a line L that passes vertically through the centre of gravity of the plate.

[Ans.image placeholder]

.

9. Find both the mean and root-mean-square values of the following functions over the given intervals:

(i) image placeholder image placeholder,

(ii) image placeholder image placeholder.

[Ans. (i) mean value = 19/3, r.m.s. value=image placeholder, (ii) mean value = 2+image placeholder, r.m.s. value = image placeholder].