# what is the difference between these two Integration over Region?

integ1 = Integrate[x^2, Element[{x, y}, Region[Disk[{0, 0}, 1]]]]
(*  ==> Pi/4 *)
integ2 = Integrate[x^2, Element[x, Region[Disk[{0, 0}, 1]]]]
(*  ==> {Pi/4, Pi/4} *)


If the region integrated over is irregular, such as a Polygon, the integ2 will give two different values.

I was wondered by the behavior of integ2. Can someone give me a hint?

In the second example, x ∈ Disk[{0, 0}, 1] (I dropped the unnecessary Region wrapper) means that x is a 2-element list. In effect, the second integral is equivalent to:

Integrate[{x, y}^2, {x, y} ∈ Disk[{0, 0}, 1]]


{π/4, π/4}

• A brief lesson! Thank you. – PureLine Oct 11 '17 at 6:57
Integrate[x^2, Element[{x, y}, Region[Disk[{0, 0}, {1, 2}]]]]
(*\[Pi]/2*)
Integrate[y^2, Element[{x, y}, Region[Disk[{0, 0}, {1, 2}]]]]
(*2 \[Pi]*)
Integrate[a^2, Element[a, Region[Disk[{0, 0}, {1, 2}]]]]
(*{\[Pi]/2,2 \[Pi]}*)


Change the integral area, and the difference maybe become obvious.