I know how to solve it by the use of integrals, but I wonder why solving it using a graph would bring incorrect result? I mean, why would it be wrong to just consider roots of these functions and solve it drawing solely on them?
Consider two functions:
$f (x) = a − x^2 $
$g(x) = x^4 − a $
For precisely which values of $a > 0$ is the area of the region bounded by the $x$-axis and the curve $y=f(x)$ bigger than the area of the region bounded by the $x$-axis and the curve $y=g(x)$?
First of all, the question is not entirely well-posed: are you intended to calculate the area along the entire infinite $x$-axis?(!) Some subinterval ($[-\sqrt{a},\sqrt{a}]$ for the $f$ and $[-a^{1/4}, a^{1/4}]$ for $g$, say)? Something else?
In any case, knowing the roots of $f$ and $g$ doesn't help you. It's like if I asked you, "I have two rectangles. One rectangle has width $n$, and the other has width $m$, but I refuse to tell you anything about the heights of the rectangles. Which rectangle has greater area?" You cannot say. Similarly you cannot say anything about the area under a curve if you just know the roots of the curve... you also need to know the heights of the curve at every $x$ to compute the area.