f(x)=-2x-1 g(x)=-1/2x+1/2 Which statements, if any, are true about these functions? I. The function f(g(x)) = x for all real x.
2 answers:
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Due to the symmetry of the paraboloid about the <em>z</em>-axis, you can treat this is a surface of revolution. Consider the curve , with
, and revolve it about the <em>y</em>-axis. The area of the resulting surface is then
But perhaps you'd like the surface integral treatment. Parameterize the surface by
with and
, where the third component follows from
Take the normal vector to the surface to be
The precise order of the partial derivatives doesn't matter, because we're ultimately interested in the magnitude of the cross product:
Then the area of the surface is
which reduces to the integral used in the surface-of-revolution setup.