Net Change Theorem
Let R (t) represent the rate at which water flows into a tank. The rate can be measured in cubic meters per second or, for example, in gallons per minute.
Then the definite integral
Instead of a water flow, we can consider any other quantity. In general case, if f (t) is the rate of change of some quantity, then the integral
This leads us to the Net Change Theorem, which states that if a quantity changes and is represented by a differentiable function, the final value equals the initial value plus the integral of the rate of change of that quantity:
The Net Change Theorem can be applied to various problems involving rate of change (such as finding volume, area, population, velocity, distance, cost, etc.)
Solved Problems
Example 1.
The engine on a boat starts at
Solution.
To estimate the fuel consumption, we use the net change theorem. This yields:
Example 2.
Suppose a fish population in a lake is increasing with a rate of
Solution.
The integral of the rate of change from
Hence, the fish population will increase by
Example 3.
A tank has a capacity of
Solution.
Let
To find the time
The roots of the equation are
Thus, the answer is
Example 4.
Suppose that
Solution.
By the net change theorem,
where
By integrating from
Thus, the final population in