The diagram shows the dimensions of a polyhedron with parallel base and top - HSC - SSCE Mathematics Extension 2 - Question 9 - 2016 - Paper 1

To find the volume of the polyhedron, we consider the geometry of the shape. The volume can be expressed using the integral of the area of the cross-section taken at height $h$ from the base to the top.

The area of the cross-section at height $h$ depends on the height and the dimensions of the base and top. According to the dimensions in the diagram, we define the area at height $h$ as:

a(h) = (h + 3) imes rac{3h}{2} + 2.

Thus, the volume $V$ can be expressed as:

V = ext{Area}(h) imes ext{height} = \int_0^4 (h + 3) imes rac{3h}{2} + 2 \, dh.

This integral clearly corresponds to option (A) from the provided choices.