Question

A solid box D is bounded by the planes x=0, x=3, y=0,

y=2, z=0, and z=1. the density of the box is given by f(x, y, z) =
2-z. Find the mass of the box.

Answer 1

To find the mass of a solid box D with given dimensions and a non-uniform density function, one must set up and evaluate a triple integral over the volume of the box using the provided density function.

Step-by-step explanation:

The student is asking how to find the mass of a solid box D with a non-uniform density function f(x, y, z) = 2 - z. To find the mass of the box, we must integrate this density function over the volume of the box. The bounds for the box are 0 ≤ x ≤ 3, 0 ≤ y ≤ 2, and 0 ≤ z ≤ 1.

Steps to Find the Mass of the Box:

Set up the triple integral for the mass: M = ∫ ∫ ∫ f(x, y, z) dV, where dV is the differential volume element.

Insert the density function and the limits into the integral: M = ∫₀³ ∫₀² ∫₀¹ (2 - z) dz dy dx.

Perform the integration first with respect to z, then y, and finally x.

After integrating, the result is the mass of the box.

By performing these calculations, the mass of the box D can be determined.