The linear density of a rod of length 4 meters is given by 𝜌(x) = 8 + 7 x measured in kilograms per meter, where x is measured in meters from one end of the rod. Find the total mass of the rod.

Answer 1

Density is just the amount of mass per unit space. In the case of a linear density, that unit space is a length. So the linear density function 𝜌(x) can be thought of as a mass per unit length, so we can write it as 𝜌(x) = dm/dx

Rearranging, we get:

𝜌(x) dx = dm

Where dm represents the mass of an infinitesimally small part of the rod, so if we summed up every single dm along the rod (i.e integrate), we get the total mass of the rod which we will call M (∫dm =M).

So integrating, we get:

∫ 𝜌(x) dx = ∫dm = M

For the limits of integration, we are summing the masses along the entire length of the rod, so we integrate from x=0 to x=4 meters

M = ∫₀⁴ 𝜌(x) dx = ∫₀⁴ 8 + 7x dx

Solving this integral, we get:

M = 8x + 3.5x^2 Ι₀⁴ = 8(4) + 3.5(4)^2 - 8(0) - 3.5(0)^2 = 32 + 3.5*16 = 88 kg

Answer 2

The density increases linearly with x. So the average density of the rod = average of the densities at the two ends = (8+7x0 + 8+7*4)/2 = (8+36)/2=22.

mass of the rod = average density x length = 22*4 = 88 kg.