圓面積

圓方程：x²+y²=r²

y=[r²-x²]^(1/2)

可先計算半圓

面積：∫^r_-rydx

=2∫^r₀[r²-x²]^(1/2)dx

代入x=rsinθ

=2∫^r₀[r²-r²sin²θ]^(1/2)d[rsinθ]

=2∫^r₀[r²cos²θ]dθ

=2∫^r₀{r²[cos(2θ)+1]/2}

=r²[sin(2θ)/2+θ]^r₀

=r²[sin(2sin^-1(x/r))/2+sin^-1(x/r)]^r₀

=r²{[π/2]-[0]}

=πr²/2

∴圓面積=πr²/2*2=πr²