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Suppose I have two functions f(y) and g(y), how do I plot the surface rotated about the axes in Maple? Assume you cannot write f(y) or g(y) in terms of x. Maple only allows g(x) and f(x)

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Do you mean you want to revolve the region between $x=f(y)$ and $x= g(y)$ around the $x$ axis? What you can do is rotate the region between $y=f(x)$ and $y=g(x)$ around the $y$ axis, and then change the orientation using transform in the plottools package. For example:

f:= y -> 2 - y^2;

g:= y -> 1 + y^2;

with(Student[Calculus1]): with(plottools):

A:= VolumeOfRevolution(f(x), g(x), x = 0 .. 1/sqrt(2), output = plot, axis = horizontal, caption = "", scaling=constrained, axes=frame):

transform((x,y,z) -> [z, y, x])(A);

enter image description here

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