The linear density of a non-homogeneous rod is expressed as ρ(x) = x² - x + 3 with unit of g/m. If the rod length is 4 m, what is the mass of the rod?

If the rod is divided into many small pieces, the total mass of the rod is the sum of the mass in each piece, which can be expressed as

When the rod is divided into infinite small pieces, the result is the exact mass value.

The definite integral definition gives

Therefore,

in which x₁ and x₂ are the start and the end of the rod. Substituting x₁ = 0, x₂ = 4, and ρ(x) = x² - x + 3 into the above formula gives