Question

A region R is bounded by the curves......

Answer 1

Why are they squared?

Answer 2

Volume = Area*height
height = dx
Area of circle = \[\huge \pi r^2 \]
If you have these concentric circles

and you are interested in the area between them; then the formula is \[\huge \pi R^2-r^2\]
Where R is the outer radius or radius of the bigger circle and r is the radius of the smaller circle.

First; write y as a function of x \[\huge x(y)=y^{\frac{1}{3}},g(y)=y^2 \]
In our case the outer radius is \[\huge x=y^{\frac{1}{3}}\] and the inner radius is \[\huge x=y^2\]the volume revolved about the y-axis would thus be \[\huge \pi\int_0^1{\left(y^{\frac{1}{3}}\right)^2 - \left(y^2 \right)^2}dy\]
\[\huge \pi\int_0^1{\left(y^{\frac{2}{3}} - y^4 \right)}dy\] so the answer is actually III

Answer 3

The smaller radii are generated by the inner function
The bigger radii are generated by the outer function.