With definite integration you can find the area under a curve (the area between the curve and the $x$ axis).
If you have a curve $f(x)$, and integrate it to get $g(x)$, you can get the area bounded by the $x$ axis, $x=a$, $x=b$, and $f(x)$, (where $a < b$), by doing $g(b) - g(a)$.
So you are getting the area under the curve up to $x=b$, and subtracting the area up to $x=a$, to get the area between $a$ and $b$.
But where does it get the area from to $a$ and $b$?, from the y axis? Because if $a=0$ then you will be taking away $0$ as $g(0)=0$, but in many curves there is often area between the curve and the $x$ axis, to the left of the $y$ axis?
Also why is area below the $x$ axis, negative area. But area to the left of the $y$ axis not negative?