Honors Calculus 252-2

Problem Set 2

to be handed in Friday January 18, 2002, in class

1. 1. Compute the mass M of a disc of unit radius lying in the x-y plane and centered at the origin with density (in polar coordinates) r(r,q) = r.
   2. Use the Law of Cosines to compute the square of the distance between a point
      (x₀,y₀) = (d,0) and a point with polar coordinates (r,q).
   3. Compute the moment of inertia I(d,0) of the disk with respect to a vertical axis of rotation through (d,0).
   4. Show that
      

      I(d,0) = I(0,0) +Md2 . (1)

      For which d is the moment of inertia minimal? Eq.(1) is an example of the parallel axis theorem.

2. For the region  defined by x²+y² £ a², x ³ 0, y ³ 0, evaluate the following integrals:
   1. 
      

      ó õ ó õ   xy dA (2)

   2. 
      

      ó õ ó õ   1   _____ Öx2+y2  dA (3)

3. 7.2.8, 7.2.11, 7.2.14, 7.2.25, 7.2.29
4. 7.3.4,iv, 7.3.5,iv, 7.3.12