Q:Is it possible to evaluate the integral of a continuous function f(x,y) over a rectangular region in the xy-plane and get different answers depending on the order of integration? Give reasons for your answer.
- LeslieLv 71 decade agoFavorite Answer
是否可能由於對 x 先積, 或對 y 先積,
No. The order of integration is insignificant for
any rectangle region.
The reason is: Fubini's Theorem (for double interal in terms of iterated integrals on rectangles), which says they are
exactly the same. The proof of this theorem is usually not
covered in the elementary calculus. But, intuitively,
this is like cutting a bread -- you can cut it in slices either
along the x-axis or y-axis, and when they are put together, it
is the same bread.Source(s): CALCULUS textbook