Exercises - Other applications

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

  

  Find its average velocity on [0,2].
  
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• 2. Find the length of the curve given by

  

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• 3. Find the length of the curve given by

  

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• 4. Find the length of the curve given by

  

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

  

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

  

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• 7. Consider the curve

  

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

  

  Find the surface area of the surface obtained by revolving this curve about the x-axis.
  
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• 9. Find the center of mass of a homogeneous half-disc of radius R.

  

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

  x² + y² ≤ z ≤ 9.

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• 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.

  

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• 12. Find the derivative F′ of the function

  

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• 13. Find the derivative F′ of the function

  

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• 14. Evaluate .
  
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• 15. Evaluate .
  
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