Finding the volume of a rectangular-based pyramid using calculus?Find the volume of a pyramid with height h and rectangular base with dimensions b and 2b.

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Andre Ferguson · Accepted Answer

Step 1Consider the pyramid having apex point at the origin and its axis coinciding with the x-axis then at a distance x from the origin, consider an elementary cuboid having small thickness dx and a rectangular cross-section of width       b    x    =            b      x        h   and length   l  x  =            2      b      x        h  .Then the volume of elementary cuboid  d  V  =  (area of rectangular cross section)  ×      (    t    h    i    c    k    n    e    s    s    )    d  V  =      b    x        l    x    d  x  =            b      x        h    ⋅            2      b      x        h    ⋅  d  x  =            2              b        2                    h      2            x    2    d  xStep 2Hence, the total volume of the pyramid  V  =  ∫  d  V  =  ∫            2              b        2                    h      2            x    2    d  xUsing the proper limits of variangle x, we get volume of complete pyramid as follows  V  =      ∫          0              h                  2              b        2                    h      2            x    2    d  x  =            2              b        2                    h      2            ∫          0              h            x    2    d  x  =            2              b        2                    h      2                  [                        x          3                3            ]              0              h        =            2              b        2                    3              h        2                  [          h      3        −    0    ]    =      2    3        b    2    h

spockmonkeyqj · Accepted Answer

Step 1So at height x above the ground the cross-section is a   y  ×  2  y rectangle, which is the base of a smaller pyramid, height   h  −  x (the top slice of the larger pyramid).Using similar triangles we see that   y  =                                                  h          −          x                                                                  h                      b and the volume of a slice of small thickness   δ  x is approximately                                                   h          −          x                                                                  h                      b  ×                                                  h          −          x                                                                  h                      2  b  ×  δ  x.Step 2Can I suggest you draw a diagram yourself - for example of a plane slice of the pyramid through the vertex and parallel to the side of length 2b, which should give you a triangular section of height h and base 2b. It doesn&#039;t in fact matter if the vertex lies symmetrically above the base - if it does the triangle will be isosceles, but do another sketch too where it isn&#039;t and see that the calculation comes out the same. If you work out the diagram for this one yourself, I&#039;m sure you will be able to work out others,You need to &quot;add&quot; these slices together using the integral      ∫          x      =      0        h              2              b        2                    h      2        ⋅  (  h  −  x      )    2    d  x

Finding the volume of a rectangular-based pyramid using calculus?

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