integration 的意義

這只是純粹巧合，因為用shell method計算圓面積的時候其integrand湊巧就是圓周公式。

請詳見http://zh.wikipedia.org/wiki/%E5%9C%86%E7%9A%84%E9%9D%A2%E7%A7%AF#.E6.B4.8B.E8.91.B1.E8.AF.81.E6.98.8E。

當然絕無定理講明integrate只含一個參數的平面圖形的周界會得出其面積公式。而事實上integrate圓面積公式只會得出一個沒有特別意義的空間數量。

(1) What is integration, please refer to definition of integration that I do not go through in details.

Good question but I do not try to explain true or not true as it is relating to concept and application.

(2) Strictly speaking integrate 2πX = πx^2 + constant. So, do you
still think integrate 圓周 得出 圓面積 ?

(3) integrate 圓面積 會得出什麼呢 = relating to volume .

Recall result of integration or differentiation on polynomial

d/dx ( x^n ) = nX^(n-1) , so Integrate X^(n-1) = x^n / n + c , c is constant

For Sphere, when volume, area, perimeter, are respectively represented by equations

V=4/3 pi x^3, where x=radius,
A= pi x^2 ,
L = 2 pi X

obviously, it has already set the relation of V, A, and L by X. so, when you applying integration on L, result will be relating to A, when applying integration to A, result will be relating to V.

you can cite another example of a solid cube of volume V, area A and edge length L respectively by formula
V= X^3
A=6X^2 where A is surface area
L=X

pls apply integrate or differentiation to V,A,L respective X, sure it shows some relationship of the result because already pre-set relation of V,A and L.

Hope it is clear.