If $f\left(x\right)+f\left(y\right)=f(x+y)\text{ and }{\int }_0^3 f\left(x\right)dx=\lambda ,$ then ${\int }_{-3}^3 f\left(x\right)dx=$

Choosing $x=0,y=0\text{, we have }f\left(0\right)+f\left(0\right)=f\left(0\right)$

$\therefore f\left(0\right)=0.$Again choosing $y=-x$ we have

$f\left(x\right)+f(-x)=f\left(0\right)=0$ as shown above.

$\therefore f(-x)=-f\left(x\right).$ Thus $f\left(x\right)$ is an odd function of $x.$

Hence ${\int }_{-3}^3 f\left(x\right)dx=0\text{ by Prop. V. }$