Solving double integral without integrating

[kasse]

From an example in my book:

Int Int (2x) dxdy over R = 0

(R is the circe x^2+(y-1)^2=1)

How can one make this conclusion without integrating?

 

[HallsofIvy]

Look at the symmetry. A circle is "symmetric about the origin": If (x,y) is in the circle then so is (-x, -y). That means that for each possible x value, you have the corresponding -x and so the "effect" of the two points will cancel out.

 

[Office_Shredder]

It's actually symmetric about (0,1), but you can draw a similar conclusion to HallsofIvy's post

 

[HallsofIvy]

Right, sorry about that! If (x. 1+y) is in the circle, so is (-x, 1- y)!