Problem 6 - Calculus Early Transcendentals

Question

In the integral for the moment \(M_{x z}\) with respect to the \(x z\) -plane of a solid, why does \(y\) appear in the integrand?

Step-by-Step Solution

Answer

Answer: The variable y appears in the integrand of the integral for the moment M_xz, because it represents the distance of a mass element from the xz-plane. This distance directly affects the magnitude of the moment contribution of that mass element, making it more likely to rotate about the y-axis as it increases further from the plane.

Step 1: Understanding moment of a solid

Moment of a solid is a measure of the tendency of the solid to rotate about an axis. In this case, we are considering the moment with respect to the xz-plane. Moment about the xz-plane involves the y-axis, and the solid is more likely to rotate about the y-axis as it increases further from the plane.

Step 2: Definition of the moment M_xz

The moment M_xz of a solid with respect to the xz-plane can be defined as the integral of the product of the mass density function (ρ(x,y,z)) and the perpendicular distance (y) of mass elements from the xz-plane, integrated over the volume (V) of the solid: \[M_{xz} = \int \int \int_V \rho(x,y,z) \cdot y \:dV\]

Step 3: Explaining the role of y in the integrand

Now let's focus on the integrand, ρ(x,y,z)⋅y, of the integral. In this expression, ρ(x,y,z) represents the mass density at a point in the solid, while y is the distance of this point from the xz-plane. The product of mass density and distance y represents the moment contribution of that particular mass element to the total moment M_xz. As the distance y increases from the xz-plane, the contribution of that mass element becomes more significant, hence it appears in the integrand. In summary, the variable y appears in the integrand of the integral for the moment M_xz of a solid with respect to the xz-plane, because it represents the distance of a mass element from the xz-plane, and it directly affects the magnitude of the moment contribution of that mass element.