球體的表面積

Question

可不可以給我證明球體表面積的公式
A=4πr^2

Answer 1

To really proof that the surface area of a sphere is not that easy to type here.
You can refer to any book that talk about surface integral.

Basically, if you are interested, it is almost always covered in text book for first year math undergraduate, especially with multivariable calculus.

Here I would present a much trickier inituitive idea. Remember that the volume of every prism is one third the base area times the height. Decompose a sphere into a very small patch on its surface area, call the area a, then we connect that to the center.

Because the patch is so small, let approximate it by a plane, then it becomes a prism. The volume is one third its area times its height. It's height is approximately the radius.

Therefore, the total volume of a sphere is the total volume of many of these small prisms, and is 4/3 πr^3 = 1/3 total surface area * radius, therefore, the total surface area is then 4πr^2.

Answer 2

其實用積分就可以，為了簡單，我們考慮一個圓形其中心點位於（０，０），而半徑是R
現在們將這個圓形rotate，使它成為球體
再考慮一小表面積，其闊度為 dx
因為這小塊的高度為 y
所以這小塊的表面積是 2πydx
Since y=Rsinθ
ds=週界．寬度
ds=(2πRsinθ)dw
dw=Rdθ
therefore
ds=(2πRsinθ)Rdθ
ds=(2πR^2．sinθ)dθ
integrate both sides fromθ=πto 0
∫ds=∫(2πR^2)sinθdθ
S=(2πR^2)∫sinθdθ
=(2πR^2)(-cosθ)
Fromθ= πto 0, we have
=(2πR^2)(1+1)
=4πR^2

Answer 3

no
3.14x半徑