Suppose R is a region in XY plane which is symmetric for all quadrant about origin(e.g. Circle centerded at origin is symmetric about origin in all Quadrants)
Now if we want to calculate the area of a symmetric region R By double Integration we can do this by calculating on first quadrant and then multiply 4. Because of region is symmetric in the all Quadrants.
But if we are asked to Integrate A function over a symmetric region R ,Then we can't do the shortcut of of finding in one Quadrant and multiplying by 4. Because Double integration is not same for all Quadrants over a symmetric region
But there are some functions which can be integrated by this method ..
So My Point is .....there are some functions which give same double integration over a symmetric region in all Quadrants and some functions Not
So my Question is How to Know which Function will satisfy the property and which will not..Is there any Condition for It????