Indefinite Integrals

The water inflow rate into a storage tank is not constant but increases over time. Initially, the pump delivers water at a rate of 5 L/min. However, the inflow rate increases by 2 L/min for each additional minute due to rising pressure or system adjustments. This scenario can be described mathematically by a linear function:

It is necessary to integrate the inflow rate function to measure the total volume of water added to the tank over time. The total water volume V(t) is obtained by performing the following integral:

When the power rule is applied to evaluate the integral, both quadratic and linear terms emerge, reflecting the changing rate of inflow:

The term t² arises due to the increasing inflow rate, demonstrating that water accumulation follows a quadratic pattern over time. The term 5t represents the steady inflow component that exists from the beginning. The constant C accounts for the initial volume of water in the tank before the inflow process begins. Since integration determines the total volume by summing the instantaneous rate of change, this equation provides a comprehensive function that describes the total amount of water in the tank at any given moment.