Integration

Integrals are used in many applications. Integrals with bounds can be used to calculate the area under a graph, integration is the opposite of differentiation and since velocity and acceleration are time derivatives of distance, we can get an expression of the distance by integrating the expression for acceleration over time.

Definition

The way integrals are used to calculate areas under a graph is by dividing the area in many smaller parts where these parts become infinitely narrow. After which these small areas are summed to get the total area under the graph. In Figure 3 such a division is shown.

As can be seen, the area divided in 15 parts follows the graph much more accurate than the division in 7 parts. When integrating, the area is divided into infinitely small parts thus the area will be exactly correct.

Standard form

Standard integrals have a similar structure consisting of a few important symbols and parts. The form of an integral is shown below:

Where $\int_{a}^{b}$ means the sum of something, in this case the y-values from $x=a$ to $x=b$, $f(x)$ is the function of which will be integrated, and $dx$ means that the function is divided in small parts with a change in x value. Sometimes the bounds $a$ and $b$ are left out, in that case the area under a graph can't be calculated since there is no start and end value.

Now is shown where integration can be used and a definition of an integral is given. See the next article about integration to learn the standard rules for solving integrals.