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**TriKri** Sure it is possible! Usually (although not always) when integrating a function f(x), it is possible to find a primitive function F(x). Then the integral of f from $\displaystyle x_1$ to $\displaystyle x_2$ will be the difference $\displaystyle F(x_2) - F(x_1)$.

Say that you instead have a function $\displaystyle f(x,\,y)$ that you want to integrate two times, one time with respect to x and then a second time with respect to y. After integrating with x you will get a primitive function, let's call that one $\displaystyle F_x,$ which is a function of x and y. The integral of f between $\displaystyle x_1$ and $\displaystyle x_2$ will be $\displaystyle F_x(x_2,\,y) - F_x(x_1,\,y)$. Now you integrate this difference with respect to y. The function $\displaystyle F_x$ appears two times in the expression, using two different x values, but integrate it once for an arbitrary value of x to get a primitive function $\displaystyle F$ of $\displaystyle F_x$, which only depends on x and y (instead of a function depending on $\displaystyle x_1$, $\displaystyle x_2$ and y). Since F is the primitive function of $\displaystyle F_x$, the primitive function of $\displaystyle F_x(x_2,\,y) - F_x(x_1,\,y)$ will be $\displaystyle F(x_2,\,y) - F(x_1,\,y)$. Now, this is the moment when you insert your limits for y, to get the integral between $\displaystyle y_1$ and $\displaystyle y_2$. After doing that, your expression will look like

$\displaystyle (F(x_2,\,y_2) - F(x_1,\,y_2)) - (F(x_2,\,y_1) - F(x_1,\,y_1)) =$

$\displaystyle = F(x_2,\,y_2) - F(x_1,\,y_2) - F(x_2,\,y_1) + F(x_1,\,y_1)$

I guess this is the expression you are looking for. Note however that this only works when you have a rectangular surface in the xy-plane with sides parallel to the x and y axes, over which you want to integrate. For other shapes, you will have to find other expressions.