Chapter 22, Problem 15P

The amount of mass transported via a pipe over a period of time can be computed as

$M={\displaystyle {\int }_{t_1}^{t_2}Q\left(t\right)c\left(t\right)dt}$

where $M=$ mass $\left(\text{mg}\right)$, $t_1=$ the initial time (min), $t_2=$ the final time (min), $Q\left(t\right)$= flow rate $\left({\text{m}}^{\text{3}}\text{/min}\right)$, and $c\left(t\right)=$ concentration $\left({\text{mg/m}}^{\text{3}}\right)$. The following functional representations define the temporal variations in flow and concentration:

$Q\left(t\right)=9+5{\cos }^2\left(0.4t\right)$

$c\left(t\right)=5e^{-0.5t}+2e^{0.15t}$

Determine the mass transported between $t_1=2$ and $t_2=8$ min with Romberg integration to a tolerance of 0.1%.