Frequently Asked Questions: The Total Change Theorem
The Total Change Theorem states that the definite integral of a rate of change of a quantity gives the total change in that quantity over a given interval. If F'(x) is the rate of change of some quantity F(x), then the total change in F(x) from x=a to x=b is given by the definite integral of F'(x) from a to b.
The Total Change Theorem is actually just a restatement or application of the **Second Part (or Part 2) of the Fundamental Theorem of Calculus (FTC)**. The FTC Part 2 states that if F is any antiderivative of f, then ∫ab f(x) dx = F(b) − F(a). If we let f(x) be the rate of change of F(x), i.e., f(x) = F'(x), then the theorem becomes ∫ab F'(x) dx = F(b) − F(a), which is the Total Change Theorem.
You use the Total Change Theorem when you know the rate at which something is changing over time or another variable, and you want to find the net amount by which the original quantity has changed over a specific interval. The formula is:
Total Change = ∫ab (Rate of Change) ⅆx
This also means that the final value of the quantity is the initial value plus the total change: F(b) = F(a) + ∫ab F'(x) ⅆx.
For example, if v(t) is the velocity of an object (rate of change of position), the integral of v(t) from t=a to t=b gives the total displacement (change in position) from time 'a' to time 'b'.
