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Math Help - [SOLVED] Derivation of formula using calculus?

  1. #1
    Super Member fardeen_gen's Avatar
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    [SOLVED] Derivation of formula using calculus?

    Derive formula of lateral(curved) surface area of cone using integral calculus?

    I checked the wikipedia link but it has no diagram for the proof. I couldn't understand the way they have divided the cones into triangles.
    Cone (geometry)/Proofs - Wikipedia, the free encyclopedia

    Can anybody help me with the proof? (It would be nice if anybody cud make the diagram)
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  2. #2
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    Hello, fardeen_gen!

    Derive formula of lateral (curved) surface area of cone using integral calculus.
    I read that proof, and don't care for it.
    I think those "triangles" are very clumsy to handle.

    I would approach it like this . . .
    Code:
            |
          r +               * (h,r)
            |           *   :
            |       *       : 
            |   *           :
        - - * - - - - - - - + --
            |               h
    We have a right triangle formed by the x-axis, y \:=\:\frac{r}{h}x, and x = h

    Rotate the triangle about the x-axis to form a cone with radius r and height h.


    The surface area is given by: . S \;=\;2\pi\int^a_b y\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx


    So we have: . S \;=\;2\pi\int^h_0\left(\frac{r}{h}x\right)\sqrt{1 + \left(\frac{r}{h}\right)^2}\,dx \;=\;\frac{2\pi r\sqrt{r^2+h^2}}{h^2}\int^h_0 x\,dx


    . . . . . . . = \;\frac{2\pi r\sqrt{r^2+h^2}}{h^2}\cdot\frac{1}{2}x^2\,\bigg]^h_0 \;=\;\frac{2\pi r\sqrt{r^2+h^2}}{h^2}\cdot \frac{1}{2}h^2


    Therefore: . S \;=\;\pi r\sqrt{r^2+h^2}

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  3. #3
    Super Member fardeen_gen's Avatar
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    It was great help!
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