Integrals

An integral is an operator that represents the ability to calculate the area which is bounded by a function.

The goal of a definite integral is to calculate the area under a curve. With it you are able to find the volumes of all the positive sections and negative sections, which you add together for an answer. An indefinite integral is an integral with no bounds, which is used to find the anti-derivative of a function.

Let us focus on definite integrals for now. Look at the following graph, it has red sections and green sections.

To calculate the definite integral of this entire function (meaning from negative infinity to positive infinity), we take the signed sum of all of these regions and add them together. A signed sum means to add the areas above the x-axis and remove the areas below the x-axis, just as its coloured above.

Popular Integral Notation¶

Integral notation has three important parts: the bounds, the function, and the variable on which we are integrating. An integral can also be either definite, when it has a lower-bound $a$ and upper bound $b$, or indefinite in which the bounds are absent.

This is an example of a definite integral:

$$\int_{a}^{b}f(x)dx$$

This is an example of an indefinite integral:

$$\int f(x)dx$$

The function shown as $f(x)$ is symbolic and could be expanded, it is meant to express any function which could be integrated.

The $dx$ represents the variable we are integrating. The $d$ is standard to mean that we are integrating the variable directly after it, which could be $dy$, $dr$, $d\theta$, $df$, or any other variable. It is important that the variable that we are integrating is actually present in our function

Definite Integrals¶

If we were to integrate the simple function $y=x$ from zero to infinity it would be denoted as such:

$$\int_{0}^{\infty} x\, dx$$

Indefinite Integrals¶

An indefinite integral is the opposite of the derivative. If we are to integrate a function $f(x)$, the output is called the anti-derivative of $f$ and can be denoted with the capital-letter version of the function, $F(x)$.

$$F(x)=\int f(x)dx$$

This is different from a definite integral because we are not calculating for a specific value but instead finding the function that, if we were to take the derivative of it, would result on the original function.

When considering an anti-derivative in the context of a problem it is important to remember that any information that is lost when taking the derivative of a function is not regained. Take the following example:

$$\begin{align} \text{Let } f(x)=2x+4 \\ \frac{d}{dx}[f(x)]=2 \\ \int 2dx=2x+c \end{align}$$

The original $f(x)$ function contains a constant value that is lost when the derivative was performed, and that information could not be re-gained when the function was integrated.