Understanding integrals

Building intuition

Integration is used in a context of finding accumulated impact on some value provided some function. It is also often explained as the method of finding the area under the curve of a function between two different points, which is virtually the same as finding accumulated impact (or net change), and below I’ll explain why.

Suppose we have a function which describes the rate of change for a specific value of $x$. As a simple example, it can be speed of a moving car dependent on time. How do we know the distance the car traveled between two time points given its function of speed? In other words, how do we find the accumulated change in the distance between two time points? If the speed is constantly changing then without using integral calculus we would only be able to get a close approximation of the distance, where the simplest would be taking some value, which is close to the average speed for the period, and multiplying it by time. However, we could get a better general approximation if we would split the whole time period of driving into parts, make approximations for each of the parts and then sum everything up. Let’s look at the graphical representation:

In this example we’ve split the whole time period between points $a$ and $b$ into equal parts and made approximations for each of these parts based on the starting speed of each split period. As a result, we come up with a set of rectangles instead of the curve. Since the distance is measured as the product of speed and time, for each period we can measure it as an area of a rectangle, and then calculate the sum of all areas as the traveled distance between points $a$ and $b$. This approach with approximation of area by dividing it into rectangles is called Riemann sum.

We can see on the graph that the function is only going through the upper left corner of each rectangle but we could have also fitted them so that the approximation would be based on the last point or the middle value of each period. Either way, approximation does not really mirror the real speed since the area under the bars may be lesser or greater than the actual area of the split. We can however minimize the error of approximation by splitting the whole area into more periods, and thus making rectangles smaller. Integration is based upon the idea of taking infinitely small splits, so that approximation approaches real function. By definition, the definite integral is the limit of the Riemann sum between two defined points when number of splits approaches infinity:

     $\displaystyle{\lim_{n \to \infty}}\sum_{i=1}^n f(x_i) \Delta x = \int_a^bf(x)dx$, where $\Delta x = \frac{b-a}{n}$ and $x_i = a+i\Delta x $

The main application of derivatives if finding the net change in some value given the function of how it changes. Definite integrals are used when the problem is restricted to some interval, for example a period of time.

The fundamental theorem of calculus

In simple words, the fundamental theorem of calculus provides connection between differentiation and integration.
Say we have a function $f(t)$ which is continuous over the interval from $a$ to $b$. If we pick some value of $x$ which belongs to the interval [$a$, $b$] then how do we define the function which represents the area between $a$ and $x$?

We already know that the area under the curve can be calculated with definite integral. The function of area under the curve $f(t)$ would look like this:

     $F(x) = \int_a^x f(t)dt$, where x in [$a$, $b$]

The fundamental theorem of calculus states that for any continuous function if we take the derivative of $F(x)$ we will get $f(x)$.

     $\frac{dF}{dx}=\frac{d}{dx}\int_a^x f(t)dt=f(x)$

Now why is this so important? It means that any continuous function has its antiderivative $F(x)$, hence there is a clear connection between differentiation and integration.

Calculating integrals

For future reference, below I include the table of the most commonly used transformations when calculating integrals.

$f(x)$	$F(x)$
$a dx$	$ax+C$
$x^n dx$	$\frac{x^{n+1}}{n+1}+C$
$e^x dx$	$e^x+C$
$a^x dx$	$\frac{a^x}{\ln (a)}+C$
$\frac{1}{x}dx$	$\ln (\lvert x \rvert)+C$

Improper integrals

Improper integrals are definite integrals that cover an unbounded area. One type of improper integrals are integrals where at least one of the endpoints is extended to infinity. When the limit exists the integral is called convergent, and when it doesn’t - it’s divergent.

Say we have the following function $f(x)=\frac{1}{x^2}$ and we want to know the whole area under the curve where $x$ is greater than 1.

And here is how we deal with it:

     $\int_1^\infty \frac{1}{x^2}dx = \displaystyle{\lim_{n \to \infty}} \int_1^n \frac{1}{x^2}dx = \displaystyle{\lim_{n \to \infty}} \left[-\frac{1}{x}\right]_ 1^n = \displaystyle{\lim_{n \to \infty}} \left[-\frac{1}{n} - (-1)\right] = \displaystyle{\lim_{n \to \infty}} \left[1 -\frac{1}{n}\right] = 1$

Since at the end we get the limit which actually exists the integral is convergent, otherwise it would be impossible to calculate the area under the curve.

Probability density function

Every random variable has its own probability density function (PDF) which describes the realtive likelihood of assuming a certain value. The absolute likelihood of a random variable assuming a certain exact value in the continuous range is in fact 0 (since there is an infinite set of possible values to begin with). The relative likelihood however indicates how much more likely the value of the variable will fall within a range of values.

Here is the plot of PDF of some random variable:

In this example we see that the relative probability of drawing a value 1.8 to 2.2 is lower than a value from 5.8 to 6.2. Also the likelihood of drawing a value greater than 9 is quite low. Actually, the area under the curve between two values on the $x$-axis constitutes the absolute probability of a variable falling in this range. It should also be clear that the probability of drawing any possible number is 1, and it corresponds to the total area under the curve of probability density function.

Knowing that integrals can be used for calculating the area under the curve they can be put to a great use when dealing with probability distributions. Here is the expression for the probability of a random variable assuming any value in terms of integrals.

     $\int_{-\infty}^\infty f(x)dx = 1$

From here the probability of $X$ assuming value between $a$ and $b$ is described like this:

     $P(a \leq X \leq b) = \int_a^b f(x)dx \leq 1$

Other than that, PDF can be used for obtaining the mean value of a continuous random variable.

     $\mu = \int_{-\infty}^\infty xf(x)dx$

One way of considering this expression is to imagine the sum of products of all possible values of $X$ and their respective probabilities.