Assume that f and g are continuous on [a,b] and
$\displaystyle \int_a^b f(x) dx > \int_a^b g(x) dx$
Does it necessarily follow that $\displaystyle f(x)>g(x)$ for all $\displaystyle x\in[a,b]$?
$\displaystyle If\ f(x)>g(x)\ for\ all\ x\ over\ an\ interval,$
$\displaystyle then\ \int{f(x)dx}>\int{g(x)dx}\ certainly,\ over\ the\ interval.$
The converse is not true however.
Similar to the graph given by running-gag,
$\displaystyle f(x)=4x\ from\ x=0\ to\ x=1$
when integrated, gives 2.
$\displaystyle g(x)=x+1$
when integrated over the same interval gives $\displaystyle \frac{3}{2}$
$\displaystyle f(x)<g(x)\ from\ x=0\ to\ x=\frac{1}{3}$
but
$\displaystyle f(x)>g(x)\ from\ x=\frac{1}{3}\ to\ x=1.$