In S. Thompson's Calculus Made Easy (pg. 31) He talks about an example which he writes, and I quote:
(4) The volume of a cylinder of radius $r$ and height $h$ is given by the formula $V = πr²h$. Find the rate of variation of volume with the radius when $r = 5.5 in.$ and $h = 20 in.$ If $r = h$, find the dimensions of the cylinder so that a change of 1 in. in radius causes a change of 400 cub. in. in the volume.
The rate of variation of $V$ with regard to $r$ is $dV/dr = 2πrh$ If $r = 5.5 in.$ and $h = 20 in.$ this becomes $690.8$. It means that a change of radius of 1 inch will cause a change of volume of $690.8$ cub. inch. This can easily be verified, for the volumes with $r = 5$ and $r = 6$ are $1570 cub. in.$ and $2260.8 cub. in.$ respectively, and $2260.8 – 1570 = 690.8$ Also, if $r = h$, $dV/dr = 2πr² = 400$ and $r = h = √(400/2π) = 7.98 in$
Forget about the second part for now. Considering only the first part, Does that mean for every consecutive increase in radius, the increase in volume is $690.8 in³$? Because if I use radii other than the ones used in the example, the increase in volume is different every time. For instance if radius is $30$ in., the volume is $56540 in³$ and radius $31$ gives volume $60530.8 in³$ the difference being $3990.8 in³$ which contradicts the statement
It means that a change of radius of 1 inch will cause a change of volume of $690.8$ cub. inch.
Is there anything I'm missing? Or am I interpreting it the wrong way? Please explain?