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Which is true about the volume or surface area of these prisms? A. The volume of B is greater than the volume of A. B. The surface area of A is greater than the surface area of B. C. The surface area of B is greater than the surface area of A. D. The volume of A is greater than the volume of B.
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To simplify this expression, we are going to combine like terms.
First, we can see that we have two terms with the variables being multiplied together,
and
. Together, these add to
, so our expression is now:
We also have two terms with ,
and
. Together, these add to
, so our expression is now:
Our final expression is xy - yz + 5.
Because I've gone ahead with trying to parameterize directly and learned the hard way that the resulting integral is large and annoying to work with, I'll propose a less direct approach.
Rather than compute the surface integral over straight away, let's close off the hemisphere with the disk
of radius 9 centered at the origin and coincident with the plane
. Then by the divergence theorem, since the region
is closed, we have
where is the interior of
.
has divergence
so the flux over the closed region is
The total flux over the closed surface is equal to the flux over its component surfaces, so we have
Parameterize by
with and
. Take the normal vector to
to be
Then the flux of across
is