Mostly, it is stated that area below x axis is negative. For an example, for y=cosx, upon integration(say from -pi to pi), we get area positive above X axis and below X axis, we get negative area, and total area is 0.
But for a circle(having portions below and above x axis), if we move from left side to right side of it (i.e. x=a to x=b), upon integration it gives us area of upper half only.(e.g. y=sqrt(1-x^2) from x=-1 to 1 gives 1.57 , which is half of the total area.
Is not the second case supposed to give same result like that of first one? What am I missing?
Update: Similar to cartesian graph, can there be a negative area in polar graph? I have an intuition that going counter clockwise gives positive area while going clockwise gives negative area, is this correct?