Trouble Following Diff. Equation: Why Evaluate B(x) in Final Expression?

In summary, the conversation discusses the theorem in calculus that states the differentiation of an integral with respect to the integral boundary is equal to the integrand evaluated at the boundary points. This is known as the Fundamental Theorem of Calculus and a more general theorem also exists. This concept is applied to explain why the integration of B(x) is still evaluated between the limits in the final expression, despite being differentiated.
  • #1
bitrex
193
0
I'm having some trouble following this equation:

[tex]\frac {d \Phi_B} {dt} = (-) \frac {d}{dx_C} \left[ \int_0^{\ell}dy \ \int_{x_C-w/2}^{x_C+w/2} dx B(x)\right] \frac {dx_C}{dt} = (-) v\ell [ B(x_C+w/2) - B(x_C-w/2)] \ [/tex]

Shouldn't the differentiation of the bracketed terms "killed" the integration of B(x)? Why is it still evaluated between the limits in the final expression? Thanks for any advice.
 
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  • #2
There is a theorem in calculus that
[tex]\frac{d}{dx} \int_0^x f(y) \, dy = f(x) [/tex]
(i.e. differentiating with respect to the integral boundary)
which you should be able to find in your calculus book.

You can prove your identity by taking any fixed point [itex]x_C - w/2 < x_0 < x_C + w/2[/itex] and write
[tex] \int_{x_C - w/2}^{x_C + w/2} dx B(x) = \int_{x_C - w/2}^{x_0} dx B(x) + \int_{x_0}^{x_C + w/2} dx B(x) = \int_{x_0}^{x_C + w/2} dx B(x) - \int_{x_0}^{x_C - w/2} dx B(x)[/tex]
 
  • #3
Thank you! That's a good theorem to know!
 
  • #4
Yes, it's called the Fundamental Theorem of Calculus!
 
  • #5
Well, I suppose if I can't recognize that, this is all kind of a lost cause!
 
  • #6
actually a more general theorem is:

[tex]\frac{d}{dx}\int^{b(x)}_{a(x)}f(s)ds=f(b(x))\frac{db}{dx}-f(a(x))\frac{da}{dx}[/tex]
 

FAQ: Trouble Following Diff. Equation: Why Evaluate B(x) in Final Expression?

1. What is the purpose of evaluating B(x) in the final expression?

Evaluating B(x) in the final expression allows us to determine the exact value of the solution to the differential equation. It helps us to find the specific function that satisfies the given equation and provides a complete solution.

2. How do you evaluate B(x) in the final expression?

The process of evaluating B(x) involves substituting the value of x into the final expression and simplifying the resulting equation. This can be done using mathematical techniques such as substitution, integration, and differentiation, depending on the complexity of the equation.

3. Can B(x) be evaluated at any value of x?

Typically, B(x) can be evaluated at any value of x as long as the given differential equation is valid for that value. However, there may be certain restrictions on the domain of x, such as avoiding division by zero or negative values under a square root.

4. Why is it important to evaluate B(x) in the final expression accurately?

Evaluating B(x) accurately is crucial because even a small error in the final expression can lead to significant discrepancies in the solution. This can result in incorrect conclusions and predictions, especially in scientific and engineering applications where precise calculations are necessary.

5. Are there any alternative methods to evaluate B(x) in the final expression?

Yes, there are alternative methods such as using numerical techniques like Euler's method or Runge-Kutta method to approximate the value of B(x) in the final expression. However, these methods may not always provide an exact solution and may require more computational effort.

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