UNIVERSITY OF TORONTO

DEPARTMENT OF MATHEMATICS

MAT 237 Y - MULTIVARIABLE CALCULUS

FALL-WINTER 2005-06

ASSIGNMENT #5. DUE ON FEBRUARY 16.

PROBLEMS

 

Do not submit your solutions without the cover page.

READ THE INSTRUCTIONS WRITTEN ON THAT PAGE.

  

1.  Show whether each of these integrals converges or diverges and find the value if it      converges.

a) 

b)          

c)           

 

2. a)  Let  S  be the region in the first quadrant bounded by

x y = 1 ,  x y = 3 ,  x 2 y 2 = 1  and  x 2 y 2 = 4 .

Find   .

b)  Let  S  be the region in the first quadrant bounded above by  x + y = 1 .

Find   .

 

Note that the function does not exist for the origin and explain how the integral can still be taken.

 

3. Let  B n ( r ) = { x R n  ≤ r }  be the sphere of radius  r , centred at the origin.

Let    be the volume of the sphere.

a)  Show that   .

b)  Rewrite    as    for some bounds  f  and  g  and with

a  dependent on  x  and  y .

c)  Evaluate the integral above to find    in terms of   .

d)  Find a closed form for   .

 

4.  Let    and let  S  be the unit square.

 

a)  Show that   .

b)  Show that the iterated integrals for the function  f  exist, but are unequal.

 

 

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