multi variable integration in maple

Question

In my research project, I needed to evaluate a multi-variable integration in maple. for example:
$\int(\int(\sqrt(x^2+y^2)dxdy\\ D: {(x^2+y^2 <= 2y) ,( y<=x ) , (x,y >0)}$

Answer 1

restart:

first := int( sqrt(x^2+y^2), x = y..sqrt(2*y-y^2) ):

first := simplify( first ) assuming y>0, y<1;

          1   /                /     (1/2)\         /        (1/2)    (1/2)\
        - - y \y ln(y) + 2 y ln\1 + 2     / - 2 y ln\(-y + 2)      + 2     /
          4                                                                 

                    (1/2)  (1/2)      (1/2)  \
        - 2 (-y + 2)      2      + 2 2      y/


second := int( first, y = 0..1);

                                16   10  (1/2)
                                -- - -- 2     
                                9    9        

Answer 2

Hint:

You can use the following formula:

int(expression, [x = a..b, y = c..d], f(x,y)