The Indefinite Integral and the Net Change

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The Fundamental Theorem of Calculus

Three Different Concepts
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The Indefinite Integral and the Net Change

Indefinite Integrals and Anti-derivatives
A Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy

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Substitution for Indefinite Integrals
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Double Integrals over General Regions

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Examples 5-7
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The Net Change Theorem

The net change theorem, which is a restatement of the FTC II from a different perspective, says that the net change in a function is the definite integral of its derivative.

$$\int_a^b f'(x)\, dx = f(b) - f(a)$$

Here, rather than a tool to evaluate an integral from $x=a$ to $x=b$, we view $f(b)-f(a)$ as the (net) change in $f$ from $x=a$ to $x=b$.

In particular, the net distance traveled from time $t_1$ to time $t_2$, which is $s(t_2)-s(t_1)$ where $s$ is the position function, is the integral of velocity. Similarly, the net change in velocity (final velocity minus initial velocity) is the integral of acceleration. We go through some of these computations in the following video.