Region $R$ is bounded by $f(x) = ax^2$, the $x$-axis, and the line $x = \frac { 1 } { a }$, where $a$ is a positive constant.

1. Sketch $R$ for three different values of $a$.

   Hint (a):

2. If $R$ is rotated about the $x$-axis, calculate the volume of the solid that is generated in terms of $a$.

   Step (b):

   $\int_0^{1/a}\pi(ax^2)^2dx$