Question

The velocity v of the flow of blood at a distance r from the central axis of an artery of radius R is given below, where k is the constant proportionality. v = k(R2 − r2) Find the average rate of flow of blood along a radius of the artery. (Use 0 and R as the limits of integration.)

Answer 1

The average rate of blood flow is given by

$\displaystyle\frac1{R-0}\int_0^Rv(r)\,\mathrm dr=\frac kR\int_0^R(R^2-r^2)\,\mathrm dr$
$=\dfrac kR\left(R^2r-\dfrac13r^3\right)\bigg|_{r=0}^{r=R}$
$=\dfrac kR\left(R^3-\dfrac13R^3\right)$
$=\dfrac{2kR^2}3$