what is statistics – Math Concepts – Introduction to Integrals

What is an Integral: Etymology
The word “Integral” is associated with the word “Integer”, which can be analyzed as in + tangere (“to touch”), which, essentially it means something that is un-touchable, whole, in a upright position.

What is an Integral: Continuity definition
Continuity of a function or Continuous Function is defined as the lack of any “hole” or “jump” between the points of the described element. A function may produce discontinuous elements. For example, the function f\left ( x \right )=\frac{1}{x} has a discontinuity when x=0 because \frac{1}{0} is not defined.

What is an Integral: Riemann Sum Method
Riemann Sum or Riemann Integral is a way to approximate Areas, Lines, Curves by dividing this region into shapes such as rectangles, squares, trapezoides. By adding together the size of region that these shapes occupies, you can approximate the size of the original region. Therefore, by increasing the number of shapes that you divide an Area, you increase the precision in your calculations about the size of the total area. Note that you must divide an Area into equal sizes either according to the Height OR the Base of the chosen shape. This method is used to approximate numerically the solution of a defined integral.

What is an Integral: The problem
There are multiple ways to explain what is an Integral.

By using geometry and the x-y Cartesian plane
—You have your Lucky Slug to walk from point A to point B in a wooden plank
—Your Lucky Slug do not walk in a straight line but it makes a zig-zag line
—You like to calculate the area under the Zig-Zag line that your Lucky Slug walked in the wooden plank.
—A way to do that is to cut this area in very tiny -infinitesimals- area pieces by drawing vertical lines from the Zig-Zag line until the bottom of your wooden plank (red lines)
—Then you make rectangles with the “red line” to fall into the middle of the rectangle.
—Then you may find the area that each rectangle occcupies
—Then, you add these areas together in order to find the total area that is defined by point A and B under the Zig-Zag line. Smaller rectangles means better accurancy about the size of the Total Area A-B.


Calculating the area under the Zig-Zag line
N=9 Rectangles exist under Zig-Zag Line that were drawn around the Red line that touches the Zig Zag Line.
—The Area size that a Rectangle occcupies is given by multiplicating its Height by its Base: Z=H*n
—The X_{0}=A=0 is equal to Start Point A which is equal to Zero.
—The X_{9}=B=1 is equal to End Point B which is equal to One.
—The Base of each Rectangle is equal to the difference between the next and previous X: \Delta X=X_{i-1}-X_{i}
—The size of the Base of every Rectangle must be equal:\Delta X_{01}=\Delta X_{12}=\Delta X_{i}
—Each Rectangle Base is also equal to the difference between the End and Start Point divided by the number of Rectangles \Delta X=n_{i}=\frac{B-A}{N}
—The Total Area E is almost equal to the summation of all Rectangle Areas: E= \sum_{i=1}f\left(  H_{i}\right ) * \Delta X
—By analyzing the formula we can get E by summing the Areas of all tiny Rectangles:
E=\sum Z_{i}=f\left( H_{1}\right ) * \Delta X+f\left( H_{2}\right ) * \Delta X+...+f\left( H_{i+1}\right ) * \Delta X


—By increasing the number of rectangles (Red Lines) to plus Infinity, then, then Rectangle Areas will get smaller and smaller, meaning increased precision about the Size of the Total Area E.
\lim_{k\rightarrow \infty }\left (  \sum_{i=1}^{k}f\left(  H_{i}\right )\Delta X\right )
—This is called also Riemann Sum Method or Riemann Integral.

—Then, the Limit function can be transformed into an Integral function:
\int_{b}^{a}f\left ( x \right )dx=\lim_{k\rightarrow \infty }\left (  \sum_{i=1}^{k}f\left(  H_{i}\right )\Delta X\right )
—The Integral Symbol \int symbolizes the Sum symbol S.


First Fundamental Theorem of Calculus
The First Fundamental Theorem of Calculus allows Definite integrals to be calculated in terms of indefinite integrals. The Defined Integral between two points / limits can be expressed as the difference between the anti-derivative function that describes the Upper Limit and the anti-derivative function that describes the Lower Limit of the Integral.

If f is continuous on the closed interval \left [ a,b \right ] and F is an anti-derivative of f, such that {F}'=f, then:
\int_{b}^{a}f\left ( x \right )dx= F(b)-F(a)

—When the Upper and Lower Limit of the Definite Integral are interchanged, then the Integration will be multiplied by negativity:
\int_{a}^{b}f\left ( x \right )dx= -\int_{b}^{a}f\left ( x \right )dx=-\left ( F(b)-F(a) \right )


—In the case of Indefinite Integral, the integration / antidifferentiation results in the Antiderivative plus a Constant:
\int f\left ( x \right )dx=F\left ( x \right )+C

Second Fundamental Theorem of Calculus
The Second Fundamental Theorem of Calculus states that we can obtain the function that is included in the Integral, the Integrand, by finding the derivative of that function or otherwise stated “differentiation undoes the result of integration”.

If f is continuous on an open interval I, if a is any point in the interval I, and if the function F is defined by the integral (anti-derivative):

F\left ( x \right )=\int_{a}^{x}f\left ( t \right )dt, then {F}'\left ( x \right )=f\left ( x \right ) where {F}'\left ( x \right ) is the derivative of F\left ( x \right )

Which also equally it can be written as: \frac{d}{dx}\int_{a}^{x}f\left ( t \right )dt= \frac{d}{dx}\left ( F\left ( x \right ) +C\right )=f\left ( x \right )

—When the Start and End point of the Definite Integral are the same one, then the result is Zero, your Lucky Slug did not move at all:
\int_{a}^{a}f\left ( x \right )dx= F(a)-F(a)=0

—You can split the Start and End point of the Definite Integral by creating an Intermediate point k:
\int_{b}^{a}f\left ( x \right )dx= \int_{k}^{a}f\left ( x \right )dx+\int_{b}^{k}f\left ( x \right )dx


—A constant c that can be any number and it exists inside the Integral, it can be taken out of the Integral as whole:
\int_{b}^{a}cf\left ( x \right )dx=c\int_{b}^{a}f\left ( x \right )dx

—For Indefinite Integrals: Given a function
\int_{b}^{a}f\left ( x \right )dx=F\left ( x \right )+c

—Integrals that include a summation or difference between its members can be splitted into separate Integrals of summation or difference:
\int_{b}^{a}f\left ( x \right )\pm  g\left ( x \right )dx=\int_{b}^{a}f\left ( x \right )dx\pm\int_{b}^{a}g\left ( x \right )dx

What is an Integral: Parts of Integral

What is an Integral: Symbols
—Closed interval \left [ a,b \right ]: It is an interval that the Upper and Lower parts of it are included in the calculations.
—Defined Integrals are the Integrals that both of their limits are defined: \int_{b}^{a}, a,b\neq \pm \infty
—Indefinite Integrals are the Integrals that have no Upper or Lower defined limits: \int
—Improper integrals or Infinite integrals are the Integrals that either their Upper or Lower limit or both are equal to plus / minus Infinity
I) \int_{+\infty}^{a}II)\int_{b}^{-\infty}III)\int_{+\infty}^{-\infty}
—Or having an integrand that becomes infinite within the limits of integration
—The Integral Symbol I\int means to Integrade or to Antidifferentiate
—The dx symbol means to integrade the function with respect to the variable x
—The Definite Integral gives the size of the Area under a Curve
—The Indefinite Integral is also known as Antiderivative and gives the general original Function because the constant term C is lost.

What is an Integral: Useful definitions / Disambiguations
—When we try to solve an Integral, by substituting its members / functions, then this is called Integration. Given the function you try to find what is equal to. In that example the size of the Total Area under the Zig-Zag line that the Lucky Slug created.
—Differentiation process (find the derivative) is the opposite of Integration process. You try to find the function while you know what this function is equal to. This is called Differentiation.
—The process of finding a function from its derivative is called anti-differentiation (indefinite integration).
—The Integration or Antidifferentiation of a function or to solve the Antiderivative (Indefinite Integral) gives the Derivative of that function plus a Constant that usually is set to zero C=0 for convenience, because its true value is not known.
—Given the original function, its Constant is known, and when it is integrated, this constant can be kept.
\int f\left ( x \right )dx=F\left ( x \right )+C \neq F\left ( x \right )+C=\int f\left ( x \right )dx+C
F\left ( x \right ) denotes the antiderivative of f\left ( x \right ) as above.
—The following equation means to differentiate f\left ( x \right ) in respect of x
f\left ( x \right ) is the antiderivative of {f}'\left ( x \right )
\frac{d}{dx}f\left ( x \right )={f}'\left ( x \right )

—Let’s say that the Line that your Lucky Slug draw as it walked can be described by the following function: f\left ( x \right )=sin(x)+3
—We know that a=0 and b=1

By using concept of Physics: The Lucky Slug paradigm
—By knowing the Velocity that your Lucky Slug travel between tiny, tiny pieces of time t, namely, between t_{a} and t_{b},
then you know what distance has traveled between that tiny tiny, time.
—But if you continue to write down the Velocity that your Lucky Slug travel between many tiny, tiny pieces of time, how can you measure the Total Distance that
your Lucky Slug has traveled?
—And by knowing the Total Distance that your Lucky Slug has traveled over time, how can you TAKE BACK its velocity?

using its x-y Cartesian plane:

By using Physics concepts:
—How can you measure an area between two points with high precision?

using its x-y Cartesian plane:

In science, infinitesimals represent all these tiny, tiny things that cannot be measured, even they can hold some mathematical, statistical, or other physical property.

So how can measure all these tiny tiny properties?

What is an Integral: The theorem
The theory of Integration states thatall these tiny, tiny things which they can represent e.g. tiny, tiny changes over time ( time=change <=> change=time ) for something, they can be added together, and therefore, providing a better understanding how something has changed over something else, as a whole e.g.

To learn about:
—How much distance your lucky slug travel a day – by writing down each day – the distance that your lucky slug has traveled over a specified timeframe, assuming that each day, this lucky slug is doing the same exact distant, with the exact same movements over time.
—what quantity of water is wasted over your old-faucet by writing down the quantity of water that is tipping through it over some tiny, tiny pieces of time.

To learn about:
—How can you describe an area under a curve

What is an Integral: Statistical Definition


Antiderivative and Indefinite Integrals