Anonymous

# 幫解一題英文微積分問題

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.

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• Leslie
Lv 7

問題:

當求一個連續函數 f(x,y)

在 xy平面的長方形區域上積分時,

是否可能由於對 x 先積, 或對 y 先積,

這積分次序不同, 而得到不同的答案值呢?

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