Math101 · Independent Ontario learning library
Math101Definite Integral
A definite integral measures signed accumulation over an interval and can be evaluated using antiderivatives.
The definite integral $\int_a^b f(x)\,dx$ is the net accumulation of $f$ from $a$ to $b$.
What it measures
If $f$ is a rate, its integral gives accumulated change. On a graph, it gives signed area: regions above the $x$-axis count positively and regions below count negatively.
Riemann-sum definition
$$
\int_a^b f(x)\,dx
=\lim_{\max\Delta x_i\to0}\sum_{i=1}^n f(x_i^*)\Delta x_i,
$$
when the limit exists. The $dx$ identifies the variable and reflects the widths being accumulated.
Worked example
Signed area versus total area
The total geometric area is $1$, not $0$. To find total area, split where the function changes sign or integrate $|f(x)|$.
Units
Integral units are output units times input units. Velocity in metres per second integrated over seconds gives displacement in metres.
