I am trying to change the order of integration of a problem and I am getting confused at a point

17: Changing Order of Integration – Valuable Vector Calculus
17: Changing Order of Integration – Valuable Vector Calculus

My math book has a question of changing order of the following integral:

$$\int_0^a\int_{\sqrt{ax-x^2}}^{\sqrt{ax}} f(x,y)\,dx\,dy.$$

So basically I have to change $dxdy$ into $dydx$ and their respective limits.

This is an example which is provided in my math textbook related to the problem. (For showing you diagram and make it easier for me to explain my confusion further.)

So in my problem, while changing the order of integration, the resultant changed integral order is coming in 3 terms (say, IOnfoot1+IOnfoot2+IOnfoot3). I1 is for OLC, I2 is for AMC and I3 is for BLCM.(Please refer to the diagram, which I shared in the form of Image)

While changing the circle, i.e. $y=\sqrt{ax-x^2}$ into form of $x$, we get $x=\frac{1}{2}\left(a\pm\sqrt{a^2-4y^2}\right)$.

I tried to solve it ahead but I stucked on a problem of deciding in ± which to choose to get higher (upper) limit and to get lower limit. But in the answersheet of my Textbook, the following is given:-

In OLC the limit of y changes into [ from y=0 to y=a/2 ] and limit of x changes into [ x= y^2/a to x= {a-sqrt(a^2-4y^2) }/2] Which means {a-sqrt(a^2-4y^2)}/2 is greater than {a+sqrt(a^2-4y^2}/2

In AMC, again we see that {a+sqrt(a^2-4y^2}/2 is taken as lower limit of x.

So, I could not understand how to decide or check which number will be greater and which number will be smaller between {a-sqrt(a^2-4y^2)}/2 is and {a+sqrt(a^2-4y^2}/2

My questions are:-

(i) How can we testify that which number can be taken as upper limit and which number can be taken as lower limit of x in the different terms while changing the order of integration, in accordance with the given question. (ii) Is there any method to test whether one algebraic expression (or number) is greater than the other in Mathematics? If yes, then please tell me.

P.S.:- Thank You So Much in advance if you willing to amswer my question. Thanks again if you edited my question to make math expressions easily understable, as I don’t know how to do it here.

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