Find the volume of the wedge in the figure by integrating the area of vertical cross‑sections.

Question

Assume that 𝑎=6,𝑏=4, and 𝑐=2.

Answer 1

This question needs the fiigure to answer it.

I am assuming the question is find the volume of a pyramid in the first octant going through the points (6,0,0), (0,4, 0), and (0,0,2).

The question asks to use cross-sections so we need to find the area of the cross section. The cross sections are right triangles so the standard equation will work.

A = 1/2 base×height

Let the height sit on the x-z plane.

h = -1/3 x+2. (found using a line through the points)

Let the length sit on the x-y plane.

l = -2/3 x+4

Multiplying the area times the width gives the volume of the infinitesimal volume.

dV = 1/2(-1/3 x+2)(-2/3 x+4) dx

Then

V = ∫⁶₀ 1/2(-1/3 x+2)(-2/3 x+4) dx = 8

We can check the answer with they pyramid formula

V = 1/3 base×height

V = 1/3 (1/2×4×2) 6 = 8