Calculus: Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method.?

In general , washers, and ∫...dx is the most straightforward .

Firstly sketch the region .
the curves y = 3√x and y = 3 meet at 3√x = 3 , so √x = 1 and x = 1
This gives the limits of x = 0 up to 1
This washer has no hole , so R = (3 -3√x) and R^2 = (9 - 18√x +9x)
the inner, small radius is 0 so r^2 = 0
the element of volume is dV = π (R^2 -r^2) dx
dV = π[ (9 - 18√x +9x)) -0 ] dx
and integrating term by term
V = ∫dV =π ∫ (9 - 18√x +9x)) -0 ] dx
V = π[9x - 18x^(3/2 ) /(3/2) +9x^2/2 ] from 0 to 1
V = π(9 - 12 + 4.5 -0 +0 -0) = 1.5 π <---- exact answer
V = 4.7124... <---- numerical answer

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Confirmation :
Using a TI-84 program, called VOLUME
ans = 4.712... as well