Question

Consider a pyramid with height h and a rectangular base with dimensions b and 2b.

(a) Cross-sections that are parallel to the base are rectangles. Find a formula for the area A(z) of the cross-section parallel to the base and z units above it.

(b) Find the volume of the pyramid by integrating the function A(z).

Answer 1

Method 1: Using Calculus

Consider the pyramid having apex point at the origin & its axis coinciding with the x-axis then at a distance xx from the origin, consider an elementary cuboid having small thickness dxdx & a rectangular cross-section of width bx=bxhbx=bxh & length lx=2bxhlx=2bxh
Then the volume of elementary cuboid

dV=(area of rectangular cross section)×(thickness)dV=(area of rectangular cross section)×(thickness)

dV=bxlxdx=bxh⋅2bxh⋅dx=2b2h2x2dxdV=bxlxdx=bxh⋅2bxh⋅dx=2b2h2x2dx

Hence, the total volume of the pyramid

V=∫dV=∫2b2h2x2dxV=∫dV=∫2b2h2x2dx

Using the proper limits of variangle xx, we get volume of complete pyramid as follows

V=∫h02b2h2x2dxV=∫0h2b2h2x2dx

=2b2h2∫h0x2dx=2b2h2∫0hx2dx

=2b2h2[x33]h0=2b2h2[x33]0h

=2b23h2[h3−0]=2b23h2[h3−0]

=23b2h=23b2h

Method 2: Using Geometry

Volume of the right pyramid with rectangular base

=13(area of rectangular base)×(vertical height)=13(area of rectangular base)×(vertical height)

=13(b⋅2b)×(h)=13(b⋅2b)×(h)

=23b2h