Avg Value Of A Function: Word Problem

[RedBarchetta]

Homework Statement

http://www.webassign.net/www20/symImages/2/1/9a952b874da3363f9e9d0b7744eaff.gif
The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is v(r), where P is the pressure difference between the ends of the vessel and η is the viscosity of the blood.

(a)Find the average velocity v_ave (with respect to r) over the interval 0 ≤ r ≤ R.

(b)Compare the average velocity v_ave with the maximum velocity v_max.(v_ave/v_max)

So I need to set the problem up, into an integral form, to find the avg velocity for part A. The integration limits for the integral should be from zero to Big R, multiplied by (1 over big R) to get our average value. Now the problem is, every thing in the equation except the (1/4) is a variable. How would I even begin to integrate this?

Thanks.

 

[dx]

The only variable is r. The rest of those symbols are constants.

 

[RedBarchetta]

Ah. Well now I've solved part A, how would go about part B?

(a) is PR^{3}/6nlR

I have V_ave, what is V max?

 

[dx]

You have a formula for v as a function of r. At what value of r is it maximum?

 

[RedBarchetta]

So if you make the equation v(R) instead of v(r), then the (R^{2}-R^{2}) = equal zero, so V_max is zero?

 

[dx]

Why did you do that? Think of it this way - the velocity at which the fluid flows depends on the distance from the center of the pipe. At a particular value of that distance, the velocity is maximum. Try drawing a picture. Plot v as a function of r. Also, first try to guess what it should be. Where do you think the water is flowing fastest? Near the center? Near the edge? Somewhere in between?