If $f$ and $g$ are continuous functions on $[0,1]$ satisfying $f(x) \geq g(x)$ for every $0 \leq x \leq 1$, and if $\int_0^1 f(x)\, dx = \int_0^1 g(x)\, dx$, then show that $f = g$.
If $f$ and $g$ are continuous functions on $[0,1]$ satisfying $f(x) \geq g(x)$ for every $0 \leq x \leq 1$, and if $\int_0^1 f(x)\, dx = \int_0^1 g(x)\, dx$, then show that $f = g$.