asked in 科學數學 · 2 decades ago

橢圓形面積公式證明??????

已知橢圓形面積公式為πab

試問如何證明

Update:

能不使用微積分證明嗎

1 Answer

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  • 2 decades ago
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    以下用積分證明

    橢圓方程式 x^2/a^2 + y^2/b^2 = 1

    令x=a*cosθ,y=b*sinθ

    則四分之一橢圓面積=∫(0~a) y*dx = ∫(π/2~0) b*sinθ*d(a*cosθ)

    =∫(π/2~0) b*sinθ*(-a*sinθ)*dθ=∫(0~π/2) ab*(sinθ)^2*dθ

    =∫(0~π/2) ab*[1-cos(2θ)]/2*dθ=ab/2*∫(0~π/2) [1-cos(2θ)]*dθ

    =ab/2*[θ-sin(2θ)] (0~π/2)=ab/2*[π/2-sin(π)] - ab/2*[0-sin(0)]=πab/4

    所以橢圓面積=4*πab/4=πab

    Source(s): 微積分
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