We use Fubini rule for the computation of double integrals and we begin by integrating in time for the $\,\, {{\partial}\over{\partial t}} \,\,$ term :

$ \displaystyle
\int_{ \displaystyle V_{j+1/2}^{n+1/2}} \, 
{{\partial W}\over{\partial t}} \, {\rm d}x \, {\rm d}t \,\, =\,\,
\int_{\displaystyle x_{j}}^{\displaystyle x_{j\!+\!1}} \,  \Bigl[ \, 
\int_ {\displaystyle t^n }^{\displaystyle t^{n\!+\!1}} \,
\, {{\partial W}\over{\partial t}}(x,\,t) \,  {\rm d}t  \Bigr] \,{\rm d}x \,\,$

$ \displaystyle \qquad  \qquad 
= \,\, \int_{\displaystyle x_{j}}^{\displaystyle x_{j\!+\!1}} \,  
\Bigl[ \, W(x,\,t^{n+1}) -  W(x,\,t^{n}) \,
\Bigr] \, \,{\rm d}x \,\,= \,\, \Delta x   \Bigl[ \,W_{j+1/2}^{n+1} -  W_{j+1/2}^{n}
\, \Bigr]  \,\, $

due to the definition (2.1.8). We proceed in an analogous way with the $\,\,
{{\partial}\over{\partial x}} \,\,$ term and begin now the Fubini procedure by integrating in space ; we have

$ \displaystyle
\int_{ \displaystyle V_{j+1/2}^{n+1/2}} \, 
{{\partial }\over{\partial x}} \bigl( a \, W \bigr)  \, {\rm d}x \, {\rm d}t \,\, =\,\,
 \int_ {\displaystyle t^n }^{\displaystyle t^{n\!+\!1}} \, 
\Bigl[ \,  \int_{\displaystyle x_{j}}^{\displaystyle x_{j\!+\!1}} \,  {{\partial
}\over{\partial x}}\,\bigl( a \, W \bigr) (x,\,t) \, {\rm d}x \, \Bigr] \, {\rm d}t
\,\,$

$ \displaystyle \qquad \qquad 
= \,\,  \int_ {\displaystyle t^n }^{\displaystyle t^{n\!+\!1}} 
\, \Bigl[ \, ( a \, W \bigr) (x_{j+1},\,t) - 