Explanation

(a)
The required volume of the segment is generated by revolving the area $ABCA$ of the circle $x^{2} + y^{2} = a^{2}$ about the $x$-axis and for the arc $BA$.
[image:8_image98.jpeg]
Here, $CA = h$
$and\ OA = a$ [given]
$\therefore OC = OA - CA = a - h$
$\therefore x$ varies from $a - h\ to\ a\$
$\therefore The\ required\ volume = \int_{a - h}^{a}{{\pi y}^{2}\ dx}$
$= \pi\int_{a - h}^{a}{\left( a^{2} - x^{2} \right)dx = \pi\left\lbrack a^{2}x - \frac{1}{3}x^{3} \right\rbrack_{a - h}^{a}}$
$= \pi\left\lbrack \left( a^{3} - \frac{1}{3}a^{3} \right) - \left\{ a^{3} - a^{2}h - \frac{1}{3}(a^{3} - {3a}^{2}h + 3{ah}^{2} - h^{3}) \right\} \right\rbrack$
$= \pi\left\lbrack a^{2}h - a^{2}h + {ah}^{2} - \frac{1}{3}h^{3} \right\rbrack = \frac{1}{3}{\pi h}^{2}(3a - h)$