School of Mathematics and Statistics
Carleton University
Math. 69.204B
ANSWERS TO TEST 4
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by interchanging
the order of integration, (that is, by using the ``other" slice).
(NOTE: This question appeared in last year's final exam. Sketch the region using the curve 
to describe R first and recall that 
means that 
.)
Solution: We can write 
, where
R is the region consisting of those (x,y) with
and 
.
The region R can be described using vertical slices as
and 
. So
where D is the region in the first quadrant bounded by the x-axis, the
y-axis and the circular arc 
with 
.
(Hint: use polar coordinates.)
Solution: In polar coordinates, the integrand is
Hence the double integral is
using spherical coordinates where T is the semi-spherical region (i.e., half-sphere) bounded below by the xy-plane and
bounded above by the surface 
with 
.
Solution: Setting 
and
we have
