• 1. [2 marks] The double integral (here a is any positive number) is equal to (c) .

  Reason: Using , we have

  

• 2. [2.5 marks] The double integral

  

  where is the disk of radius R centered at the origin: , is equal to (d) .

• (Hint: use polar coordinates.)

  Reason: In polar coordinates, the integrand is

  

  Hence the double integral is

  

• 3. [3 marks] Let D be the region bounded by the curve and the line y=x, between x=0 and x=1. Then the double integral is equal to

  (b) .

• (Hint: There are two ways to put a double integral into an iterated integral; see (4) and (5) on P. 895. We use the ``slice method'' ...)

  Reason: The description of D using vertical slices is given by

  

  In this case, the integral looks like

  

  which, when evaluated, gives

  

  On the other hand, if we use horizontal slices then D can be described by the set

  

  In this case, the integral looks like

  

  which, when evaluated, gives

  

  So you can see that, in this case, horizontal slices were easier !? It's hard to tell sometimes...

• 4. [2.5 marks] The volume of the solid that lies inside the cylinder , above the xy-plane, and below the plane z=2x+3, is given by (e) .

  Reason: Let D be the circular region in the xy-plane given by . Then the required volume is given by the double integral , since the function for . We use polar coordinates to compute this double integral. Setting and

  

  we have:

  

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