The first step is to find the volume of one slice of the pyramid. The slice is considered as a square cuboid with side x x and infinitesimally small height dh as shown in blue in the figure below, so we can affirm that the slice’s volume is d v = x 2 d h d v = x 2 d h . To continue, we need to find a relationship between x x and h h before applying integration. h h is the distance between the cuboid and the pyramid's square base. So to do that, we use Thales ‘s theorem in the triangle with the orange line segments. We get, x 2 √ a 2 √ = H − h H x a = H − h H or x = a ( H − h H ) = = − ( a H ) 2 ( − H 3 3 ) = − ( a H ) 2 ( − H 3 3 ) = a 2 H 3 3 H 2 = a 2 H 3 3 H 2 = 1 3 a 2 H
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