For the other two sides of the region, it is worth noting that x ± ct is a constant, namely x i ± ct i, where the sign is chosen appropriately. Using this, we can get the relation d x ± c d t = 0, again choosing the right sign: ∫ ∫ L 1 ( − − c 2 u x ( x , t ) d t − − u t ( x , t ) d x ) = ∫ ∫ L 1 ( c u x ( x , t ) d x + c u t ( x , t ) d t ) = c ∫ ∫ L 1 d u ( x , t ) = c u ( x i , t i ) − − c f ( x i + c t i ) .