Changing order of integration restricted by square root and circle

Question

This is not a homework (it's an exam preparation exercise set). I have an issue with changing the order of integration of the following integral:

$$\displaystyle \int_0^4 \int_{\sqrt{4x-x^{2}}}^{2\sqrt{x}} f(x,y)dydx $$

I would like to know whether it's possible to change the order of integration without dividing the area into 5 general regions in respect to Y axis as I have not been able to reduce it further.

Plot: http://www.wolframalpha.com/input/?i=plot+y+%3D+2+*+sqrt(x)+and+y+%3D+sqrt(4x+-+x%5E2)

Answer 1

Why $5$ regions? Look at the graphs:

• When $2\leqslant y\leqslant 4$, then, for each $y$, $x$ can take any value from $\frac{y^2}2$ to $4$.
• Otherwise, for each $y$, $x$ can take values from $\frac{y^2}2$ to $2-\sqrt{4-y^2}$ and from $2+\sqrt{4-y^2}$ to $4$.

Therefore, the answer is: $$\int_0^2\int_{\frac{y^2}2}^{2-\sqrt{4-y^2}}f(x,y)\,\mathrm dx\,\mathrm dy+\int_0^2\int_{2+\sqrt{4-y^2}}^4f(x,y)\,\mathrm dx\,\mathrm dy+\int_2^4\int_{\frac{y^2}2}^4f(x,y)\,\mathrm dx\,\mathrm dy.$$