## Exercises - Other applications

If you want to refer to sections of Survey while working the exercises, you can click here and it will appear in a separate full-size window. Similarly, here we offer Theory.

• 1. The instantaneous velocity of some object is given by

Find its average velocity on [0,2].

Hint

• 2. Find the length of the curve given by

• 3. Find the length of the curve given by

• 4. Find the length of the curve given by

• 5. Find the length of the curve given by parametric equations

• 6. Find the length of the curve given in polar coordinates by

• 7. Consider the curve

Find the surface area of the surface obtained by revolving this curve about the y-axis.

Hint

• 8. Consider the curve

Find the surface area of the surface obtained by revolving this curve about the x-axis.

Hint

• 9. Find the center of mass of a homogeneous half-disc of radius R.

• 10. Find the center of mass of the homogeneous solid given by the equation

x2 + y2 ≤ z ≤ 9.

• 11. Consider the ball of radius R with center at the origin. Assume that its density is given by ϱ(x) = |x|. Find the mass and the center of mass of its half (the dome) given by x ≥ 0.

• 12. Find the derivative F of the function

• 13. Find the derivative F of the function

• 14. Evaluate .

Hint

• 15. Evaluate .

Hint