Find the average rate of flow of blood - artery

[AlvaCata]

Homework Statement

The velocity v of the flow of blood at a distance r from the central axis of an artery of radius R is v=k(R^2-r^2) where K is the constant of proportionality. Find the average rate of flow of blood along a radius of the artery. (use 0 and R as the limits of integration)

Homework Equations

Average value formula in a closed interval using intergration

The Attempt at a Solution

I'm having difficulty finding what variable to integrate.
I'm using the math to find the average value, I integrate the equation in terms of r with the upper limit R and a lower limit of 0. My answer does come out as a single number containing variables 2R^2k/3. I don't know if this is the right way of doing the problem.
Thanks for your help!
AJ

 

[dynamicsolo]

It would help if you showed the actual integration you performed. Keep in mind that the artery is treated as a cylindrical section, so your differential areas are rings (annuli) of thickness dr and circumference 2(pi)r (that is, dA = 2(pi)r dr). You are not taking a one-dimensional average of a function, but one that is two-dimensional, so you must integrate v(r) · dA and divide that result by the cross-sectional area of the artery (integral of dA). This is because there is much more area contributing to the average farther away from the symmetry axis of the artery than there is close to that axis.