How to correctly calculate this integral using u substitution?

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In summary, The attempted solution for the homework statement is to find the ratio of the input flow rate, u0·AIN, to the exit flow rate, U0. First, integrate u(r) over the cross-sectional area of the pipe to find the input flow rate and the exit flow rate. Next, use the substitution, t=1-\frac{r}{R}\, to find the r value. Finally, use the equation for conservation of mass to find the ratio, Qout/Qin.
  • #1
ana111790
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Homework Statement


Calculate the following:
[PLAIN]http://img852.imageshack.us/img852/5580/integraltocalculate.jpg [Broken]

Homework Equations



u substitution where: u= (1-r/R); du = -1/R dr

The Attempt at a Solution



= umax*INT(-R*u1/7 du)
= -umax*R*(7/8)u8/7 evaluated over 0,R
= -(7/8)*R*umax [(1- R/R)8/7 - (1- 0/R)8/7]
* equals 0
**equals 1
= (7/8)*R*umax

The answer at the end of the book says it should come out to be (49/60)*Umax (no R in the term)
Can anyone help me understand what I am doing wrong?
 
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  • #2
Did you copy the problem correctly?

What you have looks fine.

Is this the complete problem, as stated?
 
  • #3
The integration is actually part of the following fluid mechanics problem:

Homework Statement


Water flows steadily through the round pipe in the figure. The entrance velocity is Uo. The exit velocity approximates turbulent flow, u = umax(1 − r/R)1/7. Determine the ratio
Uo/umax for this incompressible flow.
[PLAIN]http://img200.imageshack.us/img200/5879/fluidsproblem.jpg [Broken]

Homework Equations


Conservation of mass

The Attempt at a Solution



Qin=Qout
Vo*Ain = Aout * INT(V(r) dr)
U0*pi*r2 = pi * r2* INT(umax*(1- r/R)1/7dr)

which is where the integration I wrote earlier comes in.
The final answer should be U0/umax = 49/60 but with my calculations I get U0/umax = 7R/8.
 
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  • #4
[tex]u(r)=u_{max}\left(1-\frac{r}{R}\right)^{1/7}[/tex] is the velocity of the fluid at distance, r, from the center of a pipe with circular cross-section. This makes sense, because the fluid at the centerflows fastest, u(r) = uMAX, and flow is slowest near the pipe surface: r → R, u(r) → 0.

The fluid is incompressible, so for an input flow velocity, u0, the input flow rate, u0·AIN, is equal to the flow rate in the pipe and equal to the exit flow rate. To get the flow rate in the pipe and thus the exit flow rate, integrate u(r) over the cross-sectional area of the pipe.

[tex]u_0\cdot A_{in}=\int_0^Ru_{max}\left(1-\frac{r}{R}\right)^{1/7}{2\pi r}\,dr\ ,[/tex] where AIN = πR2.

See if this gets your desired answer.

It should.

To integrate, use the substitution, [tex]t=1-\frac{r}{R}\ [/tex] then [tex]r=R(1-t)\,.[/tex]
 
Last edited:
  • #5
I got the right answer now. Thank you so much for your help!
 

1. How do I determine the limits of integration?

The limits of integration depend on the function being integrated and the interval over which it is being integrated. To determine the limits, you can look at the graph of the function or use techniques such as substitution or integration by parts.

2. What is the best method for solving integrals?

The best method for solving integrals depends on the type of function being integrated. Some common methods include u-substitution, integration by parts, and trigonometric substitution. It is important to try different methods and choose the one that makes the integral easier to solve.

3. How do I handle improper integrals?

Improper integrals are those with one or both limits of integration being infinite or the function being integrated having a vertical asymptote within the interval. To handle improper integrals, you can split the integral into two or more parts, use a limit to approach the infinite limit, or use a change of variable to remove the vertical asymptote.

4. Can I use a calculator to solve integrals?

Yes, there are many calculators and software programs available that can solve integrals. However, it is important to understand the steps involved in solving integrals by hand to ensure the accuracy of the results.

5. How do I check my answer for an integral?

You can check your answer for an integral by differentiating your solution and comparing it to the original function being integrated. If they are equal, then your solution is correct. You can also use online integral calculators or check with your textbook or professor for the correct answer.

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