Chapter 16.2, Problem 38E

If a wire with linear density ρ(x, y, z) lies along a space curve C, its moments of inertia about the x-, y-, and z-axes are defined as

I_x = ∫_C (y² + z²) ρ(x, y, z) ds

I_y = ∫_C (x² + z²) ρ(x, y, z) ds

I_z = ∫_C (x² + y²) ρ(x, y, z) ds

Find the moments of inertia for the wire in Exercise 35.

35. (a) Write the formulas similar to Equations 4 for the center of mass ( $\bar{x}$, $\bar{y}$, $\bar{z}$) of a thin wire in the shape of a space curve C if the wire has density function ρ(x, y, z).

(b) Find the center of mass of a wire in the shape of the helix x = 2 sin t, y = 2 cos t, z = 3t, 0 ⩽ t ⩽ 2π, if the density is a constant k.