Question

We know that linear density function is the derivative of the mass function. If linear density is given by f(x) = (2x)(x^2-9)^3.find the mass function.

Answer 1

The density function is called f(x), and it's known that this function is the derivative of the mass function, m(x):

$f(x)=2x(x^2-9)^3=m^{\prime}(x)$

Then, we require solve this differential equation for "m". Then, we integrate each part of the equality:

$\int2x(x^2-9)^3=\int m^{\prime}(x)$

But we known the right hand because the integral of a derivative is the function itself:

$\int2x(x^2-9)^3=m(x)$

Let's start with the left side:

$\begin{gathered} \text{ }\int2x(x^2-9)^3=\frac{1}{4}(x^2-9)^4+c \\ \text{ } \end{gathered}$

Then the mass function is given by:

$\frac{1}{4}(x^2-9)^4+c$