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yhmjack1 2011-10-20 22:45

M2問題.....(definite integration的)

Find the area bounded by the ellipse
x^2/a^2 + y^2/b^2 = 1
ans好似係=(3.14....)ab.....
求step....

~瘋人~ 2011-10-24 00:31
x^2 / a^2 + y^2 / b^2 = 1
(bx)^2 + (ay)^2 = (ab)^2
y=+/- sqrt [b^2 - (bx/a)^2]

area =2*∫ y dx     ( upper limit = a , lower limit = -a)
         =2*∫ sqrt [b^2 - (bx/a)^2] dx
         =2b ∫ sqrt [1-(x/a)^2] dx   using substitution, x=a sin Θ
         =2b ∫ sqrt [1-(sinΘ)^2] d(a sinΘ) (upper limit = π/2 , lower limit = -π/2)
         =2ab ∫ (cosΘ)^2 dΘ
         =ab∫ (1+cos2Θ)dΘ
         =ab [Θ+(sin2Θ)/2](upper limit = π/2 , lower limit = -π/2)
         =ab(π/2 - (-π/2))
         =πab

as a reference, circle is a special case of an ellipse with area= πr^2 , since a=b=r in a circle.





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