Find the square pyramid volume as a function of aa and HH by slicing method

 

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 dv=x2dh$\text{d}v=x^2\text{d}h$. 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,

x2√a2√=H−hH

xa=H−hH

   or x=a(H−hH   )
= 

=−(

a

H

)2

(−

H

3

3

)

$$=-(\frac{a}{H}{)}^2(-\frac{H^3}{3})$$

=

a2

H

3

3

H

2

$$=\frac{a^2{H}^3}{3H^2}$$

=13a2H