Question

Given a thin rod along the x-axis over the interval [a,b], let rho(x) denote a linear density function giving the density of the rod at a point x in the interval. Then the mass of the rod is given by:___________.

Answer 1

`m = \int\limits^a_b \rho(x) \, dx`

Explanation:

The linear density is given by
`(dm)/(dx)`.

`\rho(x) = (dm)/(dx)`

`dm = \rho(x)\,dx`

Integraring both sides to get the mass,

`m = \int\!\rho(x) \, dx + C`

C is the constant of integration.

With x between a and b,

`m = \int\limits^a_b \rho(x) \, dx`