Question

The base of S is the triangular region with vertices (0, 0), (1, 0), and (0, 1). Cross-sections perpendicular to the x-axis are squares. Find the volume V of this solid.

Answer 1

the volume of the solid is V=1/6

Explanation:

The solid S has a triangular cross section in the xy-plane with sides of length L=1. The boundaries are

x=0

y=0

y= 1-x

Since each cross section perpendicular to the x axis ( parallel to the yz-plane) is a square then:

z=x

then the volume of the solid will be

V= ∫dV=∫∫∫dxdydz ∫₀¹(∫₀¹⁻ˣ dy)(∫₀ˣdz)dx = ∫₀¹ (1-x)*x dx = ∫₀¹ (x-x²) dx = [(1/2)x²-(1/3)x³] |₀¹ = 1/2 - 1/3 = 1/6

V=1/6