Let $a > 0$. Consider the region of a plane bounded by the curve $y = \sqrt { x } e ^ { - x }$, the $x$-axis, and the straight line $x = a$ which passes through the point $\mathrm{ A }( a , 0 )$, and let $V$ be the volume of the solid obtained by rotating this region once about the $x$-axis.
(1) $V$ is expressed as a function in $a$ by
$$V = - \frac { \pi } { 4 } \left\{ ( \mathbf { N } a + \mathbf { O } ) e ^ { - \mathbf { P } a } - \mathbf { Q R } \right\} .$$
(2) Suppose that the point A starts at the origin and moves along the $x$-axis in the positive direction and that its speed at $t$ seconds is $4t$. Then the rate of change of $V$ at $t$ seconds is
$$\frac { d V } { d t } = \mathbf { R } \pi t ^ { \mathbf { S } } e ^ { - \mathbf { T } t ^ { \mathbf { U } } } .$$
This rate of change is maximized at
$$t = \frac { \sqrt { \mathbf { V } } } { 4 } ,$$
and the value of $V$ at this time is
$$V = - \frac { \pi } { 8 } \left( \mathbf { W } e ^ { - \frac { \mathbf { X } } { \mathbf { Y } } } - \mathbf { Z } \right) .$$