Given two arbitary Lebesgue integrable functions we have,
$\int f dx= \int g dx$ and $f\le g \implies f=g$ a.e.
$\int g dx=\int f dx \implies \int g dx-\int f dx=0 \implies \int( g-f) dx=0$
since $g-f\ge 0$ we know $f=g$ by property of the integral.
This proof looks soild to me but I cant really convice myself considering the "area under the curve"