Darcy built the dollhouse shown. What is the volume of the first floor? What is the Volune of the attic space?
1 answer:
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By Stokes' theorem,
where
is the circular boundary of the hemisphere 
in the 
-
plane. We can parameterize the boundary via the "standard" choice of polar coordinates, setting
where
. Then the line integral is
We can check this result by evaluating the equivalent surface integral. We have
and we can parameterize by
so that
where
and 
. Then,
as expected.
where
where
We can check this result by evaluating the equivalent surface integral. We have
and we can parameterize
so that
where
as expected.