- #1
eleteroboltz
- 7
- 0
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:
[itex]
\int_0^L f(x,y) d x = \sum\limits_{i=1}^{imax} \int_{x_{i-1}}^{x_i} f(x,y) \, d x
[/itex]
Of course, for the double integration in the domain, we have:
[itex]
\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
[/itex]
If I want to do the integrals above in a integration limit different than the hole domain, we get:
[itex]
\int_0^{y_j} f(x,y) \, d y = \sum\limits_{r=1}^{j} \int_{y_{r-1}}^{y_r} f(x,y) \, dy
[/itex]
[itex]
\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
[/itex]
But what is really troubling me is the double integration, both in the same direction ([itex]\int_0^{y}\int_0^{y} \bullet \, d y \, dy[/itex]). How do I do the same separation for the integral:
[itex]
\int_0^{y}\int_0^{y2} f(x,y1) \, d y1 \, dy2 \, = \, ?
[/itex]
Note that [itex]y1[/itex] and [itex]y2[/itex] are dummy integral variables of [itex]y[/itex].
please guys, help me.
Thanks in advance
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:
[itex]
\int_0^L f(x,y) d x = \sum\limits_{i=1}^{imax} \int_{x_{i-1}}^{x_i} f(x,y) \, d x
[/itex]
Of course, for the double integration in the domain, we have:
[itex]
\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
[/itex]
If I want to do the integrals above in a integration limit different than the hole domain, we get:
[itex]
\int_0^{y_j} f(x,y) \, d y = \sum\limits_{r=1}^{j} \int_{y_{r-1}}^{y_r} f(x,y) \, dy
[/itex]
[itex]
\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
[/itex]
But what is really troubling me is the double integration, both in the same direction ([itex]\int_0^{y}\int_0^{y} \bullet \, d y \, dy[/itex]). How do I do the same separation for the integral:
[itex]
\int_0^{y}\int_0^{y2} f(x,y1) \, d y1 \, dy2 \, = \, ?
[/itex]
Note that [itex]y1[/itex] and [itex]y2[/itex] are dummy integral variables of [itex]y[/itex].
please guys, help me.
Thanks in advance