Attempt $1$: $4$ attempts remaining.

The figure below shows $f.$

$($Click on the graph for a larger version.$)$

If $F^{\text{'}}=f$ and $F(0)=0,$ find $F(b)$ for $b=1,2,3,4,5,6,$ and fill these values in the following table.

$\backslash$table$[[$b$,1,2,3,4,5,6],[F(b),$a$,0,0,0,,]]$