Homework #11

Do all double integrals below
entirely by hand showing all the steps, except where noted otherwise. 
You may, if you wish, check your calculations with Maple's "int" command.

14.4	4. Area of a petal. (Integration in polar coordinates.)
	   Sketch the region in the (r,theta) parameter plane
	   and the region in the (x,y) measuring plane.

14.9/15.5 Integration over parametrized surfaces.
	   X. 
	   (a) Devise a parametrization of the upper (z>=0) hemisphere 
	       of radius R (centered at the origin).
	   (b) Calculate the jacobian of your parametrization, and then 
               use your parametrization to compute the area of the surface.
	   (c) If the hemisphere is made of copper (8.96 g/cm^3), it's radius
	       is 5m, and it is 1mm thick (much less than R), compute it's 
	       (approximate) mass.
	   (d) Where is the center of mass? It must be on the axis of symmetry,
	       but how far up? Guess! Explain your reasoning.
	   (e) Compute the location of the center of mass.
	       Are you surprised by the answer?

As with the double integrals, do all the triple integrals below
entirely by hand showing all the steps, except where noted otherwise. 
You may, if you wish, check your calculations with Maple.

14.6	4. Integral over a box. 

 	X. (a) Figure out how to specify the region of Problem 14.3.42, p907.
	       Use region3d to verify that you've
	       got it right.
           (b) Evaluate the volume of the region.