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Establish the prismoid formula

by RoRi

Given a solid whose cross sections have area ax^2 + bx + c for x \in [0,h]. Compute the volume of this solid in terms of B_1, M, B_2, the cross sections corresponding to x= 0, \ x = \frac{h}{2}, \ x = h, respectively.

Since the area of the cross section at x is given by ax^2 + bx + c, we have

    \begin{align*} B_1 &= c \\ M &= \frac{ah^2}{4} + \frac{bh}{2} + c \\ B_2 &= ah^2 + bh + c. \end{align*}

Then, we compute the volume,

    \begin{align*} V &= \int_0^h (ax^2 + bx + c) \, dx \\ &= \frac{ah^3}{3} + \frac{bh^2}{2} + ch \\ &= \frac{h}{6} (2ah^2 + 3bh + 6c) \\ &= \frac{h}{6} (B_1 + 4M + B_2). \end{align*}

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