Answer
Sometimes true.
Work Step by Step
As area should be a positive number, the correct expression would be $A=\int_a^b|f(x)-g(x)|dx$ which means that the expression given in the exercise is true when $f(x)-g(x)\geq0$.
If $f(x)-g(x)\lt0$, we have $A=\int_a^b[g(x)-f(x)]dx$. If the two functions intersect in the interval $[a,b]$, we need to integrate different regions and make sure the difference of the two functions is positive in each region.
