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

[bitrex]

I'm having some trouble following this equation:

$$\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)] \$$

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.

 

[CompuChip]

There is a theorem in calculus that
$$\frac{d}{dx} \int_0^x f(y) \, dy = f(x)$$
(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 $x_C - w/2 < x_0 < x_C + w/2$ and write
$$\int_{x_C - w/2}^{x_C + w/2} dx B(x) = \int_{x_C - w/2}^{x_0} dx B(x) + \int_{x_0}^{x_C + w/2} dx B(x) = \int_{x_0}^{x_C + w/2} dx B(x) - \int_{x_0}^{x_C - w/2} dx B(x)$$

 

[bitrex]

Thank you! That's a good theorem to know!

 

[HallsofIvy]

Yes, it's called the Fundamental Theorem of Calculus!

 

[bitrex]

Well, I suppose if I can't recognize that, this is all kind of a lost cause!

 

[elibj123]

actually a more general theorem is:

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