Navigation: Wiki Home > Skills Index > Indefinite Integrals and Net Change
This concept page explores the relationship between antiderivatives, indefinite integrals, and the net change theorem.
The indefinite integral of $f(x)$ is:
$$\int f(x)\,dx = F(x) + C$$
where $F$ is any antiderivative of $f$ (meaning $F'(x) = f(x)$), and $C$ is an arbitrary constant.
If $F'(x) = f(x)$ is continuous on $[a, b]$, then:
$$\int_a^b f(x)\,dx = F(b) - F(a)$$
This is the Fundamental Theorem of Calculus, Part 2.
The definite integral $\int_a^b f(x)\,dx$ gives the net change in $F$ over the interval $[a, b]$.
| If $f(x)$ represents... | Then $\int_a^b f(x)\,dx$ gives... |
|---|---|
| Velocity $v(t)$ | Displacement (net change in position) |
| Rate of population change $P'(t)$ | Net change in population |
| Marginal cost $C'(x)$ | Net change in total cost |
$$\text{Displacement} = \int_a^b v(t)\,dt$$
This can be positive, negative, or zero.
$$\text{Total Distance} = \int_a^b \vert v(t)\vert \,dt$$
This is always non-negative and accounts for all movement.
Example: If you walk 5 m east then 3 m west: