In the figure, the equation of the circle is (x^2)+(y^2)=r^2. Show that the area of the circle is πr^2 by definite integration.
圖片參考：http://img141.imageshack.us/img141/8900/53236659.jpg
definite integration
2011-03-11 4:09 am
回答 (2)
2011-03-11 4:27 am
✔ 最佳答案
Let x = rcosθdx = -rsinθdθ; x = 0, θ = π/2 ; x = r, π = 0
The area of the circle
= 4 ∫ f(x) dx [0,r]
= 4 ∫ √(r^2 - x^2) dx [0,r]
= 4 ∫ √(r^2 - r^2cos^2θ) (-rsinθ) dθ [π/2,0]
= 4r^2 ∫ √(1 - cos^2θ) (sinθ) dθ [0,π/2]
= 4r^2 ∫ (sin^2θ) dθ [0,π/2]
= 2r^2 ∫ (1 - cos2θ) dθ [0,π/2]
= πr^2 - 2r^2 ∫ cos2θ dθ [0,π/2]
= πr^2 - r^2 sin2θ [0,π/2]
= πr^2
2011-03-12 3:16 am
將 2 π r 積分
= π r^2
= π r^2
收錄日期: 2021-04-20 00:15:27
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20110310000051KK01029