Chapter 1, Problem 1.67AP

A rod extending between x = 0 and x = 14.0 cm has uniform cross-sectional area A = 9.00 cm². Its density increases steadily between its ends from 2.70 g/cm³ to 19.3 g/cm³. (a) Identify the constants B and C required in the expression ρ = B + Cx to describe the variable density. (b) The mass of the rod is given by

$m={\displaystyle \underset{\text{all material}}{\int }\rho dV}={\displaystyle \underset{\text{all x}}{\int }\rho Adx={\displaystyle {\int }_0^{\text{14}\text{.0 cm}}(B+Cx)(9.00{\text{cm}}^{\text{2}})}}dx$

Carry out the integration to find the mass of the rod.