圓周

圓方程：x²+y²=r²

2x+2y[dy/dx]=0

dy/dx=-x/y

可先計算半圓

dy/dx=-x/(r²-x²)^(1/2)

[dy/dx]²=x²/(r²-x²)   ----- (1)

計算圓周可考慮

[△s]²=[△x]²+[△y]²

[△s/△x]²=1+[△y/△x]²

[ds/dx]²=1+[dy/dx]²

[ds/dx]={1+[dy/dx]²}^(1/2)

∫^r_-r[ds/dx]dx=∫^r_-r{1+[dy/dx]²}^(1/2)dx

代入(1)

s=∫^r_-r{1+[x²/(r²-x²)]}^(1/2)dx

代入x²/(r²-x²)=tan²θ

x²=(r²-x²)tan²θ

x²[1+tan²θ]=r²tan²θ

x=rtanθcosθ=rsinθ

∴s=∫^r_-rsecθd[rsinθ]

s=∫^r_-rrdθ

s=[rθ]^r_-r=[rsin^-1x/r]^r_-r

s=r{[π/2]-[-π/2]}=πr

∴圓周=πr*2=2πr