solid of revolution

disc integration is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution
if the functions rotate around the $\text{[math]}$ axis, the formula is
$\text{[math]}$ if the functions rotate around the $\text{[math]}$ axis, the formula is
$\text{[math]}$

we start with a simple example, consider the function $\text{[math]}$
$\text{[math]}$
if we were to rotate this function around the $\text{[math]}$ axis we would get a Right Circular Cone:
$\text{[math]}$
to find the volume of this cone we use the disc integration formula, in this case $\text{[math]}$
assume the function stopped at $\text{[math]}$ and started at $\text{[math]}$ so we find the volume in the interval $\text{[math]}$
$\text{[math]}$

a circle is defined as $\text{[math]}$ , consider the circle $\text{[math]}$
$\text{[math]}$
if we were to rotate the circle $\text{[math]}$ around the $\text{[math]}$ axis we would get a ball
the upright quarter of a circle is defined as $\text{[math]}$
consider the quarter of that circle whose formula is $\text{[math]}$
$\text{[math]}$
if we were to rotate this around the $\text{[math]}$ axis we would get half a ball and since this is easier to find the volume of we just find the volume using this quarter and multiply by 2 because it generates half a ball
$\text{[math]}$
in this case $\text{[math]}$ , we find the volume using the formula:
$\text{[math]}$
so the volume of the ball is $\text{[math]}$

find the volume of produced by rotating the function $\text{[math]}$ around the $\text{[math]}$ axis in the interval $\text{[math]}$
$\text{[math]}$
$\text{[math]}$