Because you are working in polar coordinates
Blood flows through an artery of radius R. At a distance r from the central axis of the artery, the speed of the blood flow is given by S(r)=k(Rē - rē). Show that the average speed of the blood flow is one-half the maximum speed.
My problem is that I can't prove that -- I keep proving that it is two-thirds the maximum speed. Can anyone tell me where I am going wrong?
The maxiumum speed is kRē at r = 0 and the function reaches zero when r = R.
Where did I go wrong? I just can't see it. Thanks for the help!
1. First of all, how do you know that solving this problem requires the use of polar coordinates? (I warned you that this would be involved ) The example problems in the book that use this equation don't involve polar coordinates, at least not by name, so if you could explain the your mental process in addressing the problem, that would be wonderful.
2. Second, where does that extra r here: r
3. Third, how do you integrate an expression twice? Do you integrate the expression and then combine the limits of integration? a = 0, b = 2piR?
I know this is a lot to ask, but you guys give more thorough explanations that what my teacher can provide and I want to understand what I am doing.
I used polar coordinates because S is given as an expression of radius r
If you do not know about polar coordinates maybe this link could help you
Polar coordinate system - Wikipedia, the free encyclopedia
I have also performed a sketch (see below)
A point M is given by 2 coordinates and
Starting from M, when doing a little displacement along point M is sliding along
When doing a little dispalcement point M is sliding along
The elementary surface dS (which is dx dy in cartesian coordinates) is therefore
This is why there is an "extra r" in the expression of the integral
For instance if you want to compute the area of the disk
Integrating S(r) over the disk gives
S being only an expression with r, the double integral can be split into 2 parts
I don't know if my explanations are useful ... and understandable
You have been more than helpful and very patient , and your solution certainly works beautifully (I went ahead and read the chapter on integrals of polar coordinates and they certainly seem to offer the most straightforward solution to this problem). However, if you will bear with me a little longer, is there a way to work this problem that does not involve these more advanced topics?
Running-gag, thanks so much for your dedication to helping me with this problem. You really gave me a lot to think about and I broadened my mathmatical horizons.
Skeeter, thanks for your input as well. I am always hesitant to think that there is something wrong with the origianl problem, but in this case it looks like there may be, at least in the context of the lesson it is given in.
I think that it looks like I need to bring this problem to the teacher's attention, since it does not seem to work as a Calc II level problem.
Once again, thank you for all of your help and time!