$a>0$

Let $n\le a<n+1, n\in W$

Then,

$a=\left[a\right]+\left\{a\right\}$

Here, $\left[a\right]=n$
Now,

${\int }_0^a{e}^{x-\left[x\right]}dx=10e-9$

$\Rightarrow {\int }_0^n{e}^{\left\{x\right\}}dx+{\int }_n^a{e}^{x-\left[x\right]}dx=10e-9$

$\Rightarrow n{\int }_0^1{e}^{x}dx+{\int }_n^a{e}^{x-n}dx=10e-9$

$\Rightarrow n\left(e-1\right)+\left(e^{a-n}-1\right)=10e-9$

$\Rightarrow ne-n+e^{a-n}-1=10e-9$

$\Rightarrow ne-n-1+e^a{e}^{-n}=10e-9$

On comparing, we get

$n=10$

and,

$-n-1+e^a{e}^{-n}=-9$

$\Rightarrow -10-1+e^{a-10}=-9$

$\Rightarrow e^{a-10}=2$

$\Rightarrow \left(a-10\right){\log }_e\left(e\right)={\log }_e\left(2\right)$

$\Rightarrow a=\left(10+{\log }_{e}2\right)$