1. The problem involves understanding and performing a mass balance integration, which is typically used in engineering and physics to relate the mass flow rates and accumulation of a substance within a system. 2. A general mass balance equation can be written as $$\frac{dM}{dt} = \dot{m}_{in} - \dot{m}_{out} + r$$ where: - $M$ is the mass inside the system, - $\dot{m}_{in}$ is the mass flow rate in, - $\dot{m}_{out}$ is the mass flow rate out, - $r$ is the rate of mass generation within the system (could be zero if none). 3. If you want to integrate this equation over time from $t=0$ to $t=T$, you get: $$M(T) - M(0) = \int_0^T (\dot{m}_{in} - \dot{m}_{out} + r) dt$$ 4. This integral tells us the change in mass inside the system over the time interval from 0 to $T$ given the difference in the mass flow rates and generation over that time. 5. The integration step is just the accumulation of net mass flow over time. 6. This approach helps in solving problems involving accumulation or depletion of mass in tanks, reactors, or other systems. Final answer: $$M(T) = M(0) + \int_0^T (\dot{m}_{in} - \dot{m}_{out} + r) dt$$