how to find area enclosed inside a parabola? 
In the image above, we used $y=x^2$ in order to take the area below the parabola. Why is the equation giving us the area below the parabola instead of above it? What if we want the area above the curve?
 A: The integration $\int$ operation is actually likened to summing a number of rectangles that are fitted from the x-axis to the curve in a process called Riemann Sums. The $dx$ in the integral represents an infinitesimal width of a rectangle of height $f(x)$ which is $x^2$ in your case. The integration process is actually the limiting process of a thing called Riemann Sums. 
There is a nice basic explanation and graphic here of Riemann Sums. It can be proven that the Riemann sum approximation converges to the actual area under the curve in the limit.
For the area above the curve, if you have no other conditions then the area above any interval on a parabola is infinite. You would need to define a boundary to integrate below and subtract the area of the parabola to find areas above the parabola. 
For instance, on the right hand side of your diagram you would find the area between $y=x$ and $y=x^2$ from O to M as $\int^1_0 x-x^2 dx$ which is above the parabola. You almost always find areas with respect to the x-axis. Otherwise you can swap the subject of your equation to x to calculate the area between the parabola and the y-axis, so instead of $y=x^2$ you have $x=\sqrt y$ and integrate w.r.t $y$.
