Chapter 4.4, Problem 71E

Life science: Poiseuille’s Law. The flow of blood in a blood vessel is faster toward the center of the vessel and slower toward the outside. The speed of the blood is given by

$V=\frac{p}{4Lv}\left(R^2-r^2\right)$, where R is the radius of the blood vessel, r is the distance of the blood from the center of the vessel, and p, v, and L are physical constants related to the pressure and viscosity of the blood and the length of the blood vessel. If R is constant, we can regard V as a function of r:

$V\left(r\right)=\frac{p}{4Lv}\left(R^2-r^2\right)$.

$Q=\frac{\pi p{R}^4}{8Lv}$

The total blood flow, Q, is given by

$Q={\displaystyle {\int }_0^{R}2\pi \cdot V\left(r\right)\cdot dr}$.

Find Q.