Integral Separation for PhD Thesis Formulation

[eleteroboltz]

Hey guys,

I am working on my PhD thesis formulation and I got to a doubt. I need to do some integral separations, for the mesh attached, of the form:

$\int_0^L f(x,y) d x = \sum\limits_{i=1}^{imax} \int_{x_{i-1}}^{x_i} f(x,y) \, d x$

Of course, for the double integration in the domain, we have:

$\int_0^H\int_0^L f(x,y) d x \, dy = \sum\limits_{j=1}^{jmax} \sum\limits_{i=1}^{imax} \int_{y_{j-1}}^{y_j} \int_{x_{i-1}}^{x_i} f(x,y) \, dx \, dy$

If I want to do the integrals above in a integration limit different than the hole domain, we get:

$\int_0^{y_j} f(x,y) \, d y = \sum\limits_{r=1}^{j} \int_{y_{r-1}}^{y_r} f(x,y) \, dy$

$\int_0^{y_j}\int_0^{x_i} f(x,y) \, d x \, dy = \sum\limits_{r=1}^{j} \sum\limits_{q=1}^{i} \int_{y_{r-1}}^{y_r} \int_{x_{q-1}}^{x_q} f(x,y) \, dx \, dy$

But what is really troubling me is the double integration, both in the same direction ($\int_0^{y}\int_0^{y} \bullet \, d y \, dy$). How do I do the same separation for the integral:

$\int_0^{y}\int_0^{y2} f(x,y1) \, d y1 \, dy2 \, = \, ?$

Note that $y1$ and $y2$ are dummy integral variables of $y$.

please guys, help me.
Thanks in advance

 

[eleteroboltz]

OK,
After thinking a lot about it, I got the solution

$\int_0^{y_i}\int_0^{\eta} f(x,\xi) \, d\xi \, d\eta \, = \, \sum\limits_{s=1}^j\sum\limits_{r=1}^{s-1} (y_s-y_{s-1}) \, \int_{\eta_{r-1}}^{\eta_r}f(x,\xi) \, d\xi\ \, + \, \sum\limits_{s=1}^j \int_{y_{s-1}}^{y_s} \int_{\eta_{s-1}}^{\eta} f(x,\xi) \, d \xi \, d \eta$

I attached the derivation of the expression above.

Cheers