Net Change Theorem

∫ from a to b f'(x)dx = f(b) - f(a)

The integral of a rate of change gives the net change.

Applications:

If v(t) = rate of water flowing in (gallons/min):

∫ from 0 to 10 v(t)dt = total gallons added in 10 minutes.

If P'(t) = rate of population change (people/year):

∫ from 2020 to 2025 P'(t)dt = net population change over 5 years.

If C'(x) = marginal cost (dollars per unit):

∫ from 100 to 200 C'(x)dx = additional cost to produce units 101-200.

AP exam context: These problems come with real-world scenarios. Read carefully to understand what the rate represents, integrate it, and interpret the result in context with correct units.

Reference:

Wikipedia: FTC

https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus