how to sketch the region of integration and change the order of integration? ∫(0 to a.^2) ∫ (sqrt(x) to a f(x,y) dydx here a is >0?

The bounds of integration is given by
y = √x to y = a with x in [0, a^2].

Sketching this, we can rewrite this as
x = 0 to x = y^2 with y in [0, a].

So, the integral equals
∫(y = 0 to a) ∫(x = 0 to y^2) f(x, y) dx dy.

I hope this helps!

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