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[수학 정보] 적분을 이용하지 않은 원뿔의 부피 증명
리프 2020.07.31 05:45 조회 1743
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페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

밑면의 넓이가 1, 높이가 1인 원뿔의 부피를 V라고 합시다. (계산의 편의를 위해)

이때, 주어진 뿔에서 밑면으로부터 \Delta h만큼 떨어진 평행한 평면으로 원뿔을 잘라 만든 원뿔대를 생각해봅시다.

원뿔대의 부피를 구해보면 (1-(1-\Delta h)^3)V임을 알 수 있습니다.

한편, \Delta h\rightarrow 0으로 보내면 원뿔대는 원기둥으로 근사할 수 있고, 부피가 \Delta h가 됨을 알 수 있습니다.

또한, (1-(1-\Delta h)^3)V\simeq 3\Delta hV가 되므로 V=1/3이 됩니다. 

 

참고) 밑면의 넓이를 S, 높이를 h라 놓고 동일하게 식을 정리해보면 우리가 알고 있는 원뿔의 부피 공식을 얻을 수 있습니다.

참고2) 이와 같은 방식으로 n차원 뿔형 도형의 부피를 쉽게 알 수 있습니다.

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  •  
    리프 Lv.7 2020.07.31 05:47

    사실 적분을 안 썼긴 했지만 극한의 개념을 쓰는건 마찬가지네요 ㅋㅋ

    다른 증명법으로는 카빌리에리의 원리를 이용하여 원뿔의 부피와 동일한 정사각뿔을 만든 후, 정육면체 부피의 1/3임을 보이는 방법이 있습니다. (영어 위키백과에 원뿔 검색하면 나옵니다)

    댓글 작성하기 좋아요0 댓글수1
    •  
      mwryan Lv.6 2020.09.01 18:30

      흑 극한 안 쓰는 줄 알고 매우 놀라서 들어왔는데... 역시 원 들어가는 거는 다 그렇게 해야 되는 거군요...

      좋아요0
  •  
    황금 열쇠 Lv.7 2020.07.31 17:36

    오

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  •  
    아인수타인 Lv.12 2020.07.31 22:52

    카발리에리 원리로도 가능하긴 한데 이것도 적분의 일종이라...

    댓글 작성하기 좋아요0 댓글수0
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