Fluid is flowing in a tube that has a radius of 3 centimeters. Water is flowing through a circular cross section at a rate of (9-r^2) cm/s, where r is the distance from the center of the cross section. What is the total amount of water that flows through the cross section in 4 seconds?

Answer 1

at a radius r, the volume flowing through per second is

dv = 2πrh dr
where h is the height of the cylinder. That height is the rate of flow per second, so
dv = 2πr(9-r^2) dr
dv/dr = 2πr(9-r^2)

Now add up for all the thin cylinders of thickness dr, and

v = ∫[0,3] 2πr(9-r^2) dr = 81π/2 cm^3

This problem seems kind of hard to understand, but consider a stack of nested cylinders of varying heights, where the inner cylinders are taller than the outer ones, where the water flows more slowly. The area of a circular ring of radius r and width dr is just 2πr dr.