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Math Help - Volume of a Pyramid

  1. #1
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    Volume of a Pyramid

    Use calculus to find the volume of a pyramid with height h and rectangular base with dimensions b and 2b.



    I don't know how to start this question. I've tried finding the integral from 0 to h of (b)(2b)db, but that doesn't seem to work.
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  2. #2
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    Take a horizontal cross-section of the pyramid. Let s be the smaller side of the cross-section and let x be the height from the cross-section to the base of the pyramid. Due to similarity, the longer side of the cross-section must have length 2s and the area will be 2s^2. Therefore, the volume of the pyramid will be

    V=\int_{0}^{h}2s^2dx.

    To find s in terms of x, use the similarity of triangles (vertically 'slice' the pyramid in half and examine the cross-section).
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  3. #3
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    Place this imaginary pyramid on a co-ordinate system, with the apex of the pyramid at the origin. The sloping side of the pyramid forms, well a slope, with points (0,0) and (h, b) (if we use 2b, then half of its height above the x-axis will be b). Therefore an equation of the line for this pyramid would be y=\frac{b}{h}x, and the area of a cross section of this pyramid would be A(x)=\left(\frac{2b}{h}x\right)\left(b\right)\Righ  tarrow A(x)=\left(\frac{2b^{2}x}{h}\right)

    It is then simply a matter of integrating the area function using 0 and h as limits of integration to find the total volume of the pyramid.
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