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The volume of the solid obtained by rotating the region bounded by $x=3 y^2, \ y = 1, \ x = 0,$ about the $y$ -axis can be computed using the method of disks via an integral

$\displaystyle V = \int_a^b$

with limits of integration $a =$ and $b =$ .

The volume of this solid can also be computed using cylindrical shells via an integral

$\displaystyle V = \int_\alpha^\beta$

with limits of integration $\alpha =$ and $\beta =$ .

In either case, the volume is $V =$ cubic units.