Why does integrating a function within a limit of a to b give you the area of the graph from a to b?
There are various ways to construct the integral. Typically the integral of a continuous function $\displaystyle f:[a,b] \rightarrow \mathbb{R}$ is constructed as the limit of a sequence of Riemann sums.
$\displaystyle \int_a^bf(x)dx$ can also be interpreted as $\displaystyle \lim_{n\to\infty}\sum_{i=1}^nf(x_i)\cdot\Delta_ix$ where $\displaystyle f(x_i)$ represents the height of $\displaystyle f$ at some value $\displaystyle x_i$ on the closed interval $\displaystyle [a,b]$, with $\displaystyle x_1=a$ and $\displaystyle x_n=b$. $\displaystyle \Delta_ix=x_i-x_{i-1}$ represents each distinct subinterval on $\displaystyle [a,b]$ (which can be interpreted as width).
Now, with the technical stuff out of the way, you can see that $\displaystyle \lim_{n\to\infty}\sum_{i=1}^nf(x_i)\cdot\Delta_ix$ is just the sum of the area of a bunch of rectangles as the width of these rectangles tends to zero.
$\displaystyle A=w\cdot{h}$, in the riemann sum, you can see that $\displaystyle w=\Delta_ix$, and height is $\displaystyle f(x_i)$.
The sum of the area of a bunch of rectangles is an area.
the $\displaystyle \int$ symbol has stretched out over the years since newton's day but it used to look more like an $\displaystyle S$. stading for sum.
NOTE. I choose a lack of precision here in an attempt to explain the definition of the riemann integral in an intuitive way. The precise definition is
$\displaystyle f$ continuous: if
$\displaystyle |\sum_{i=1}^nf(\xi_i)\cdot\Delta_ix-A|<\epsilon$ whenever $\displaystyle ||\Delta||<\delta$.
then we say that $\displaystyle A=\int_a^bf(x)dx$
$\displaystyle \xi_i$ being any number on each $\displaystyle \Delta_ix$, and $\displaystyle ||\Delta||$ being the largest subinterval in the partitioned subdivision from $\displaystyle a$ to $\displaystyle b$.
For example if I integrate this function
$\displaystyle \int_0^2 x^2dx$
Integrating that will give you the area under the function $\displaystyle y=x^2$ from 0 to 2.
Why?
@shawsend
I can't get that. I'm not a university student, just a 14 year old kid that finished the calculus part of Kumon math curriculum.
I did not know that. Well, then. It is very good that you have come to where you are at such a young age. When you are my age you will have achieved much in the realm of mathematics.I can't get that. I'm not a university student, just a 14 year old kid that finished the calculus part of Kumon math curriculum
Maybe, if I say to you that, the way in which books are set up tend to favor practical use over conceptual understanding. Math books today are designed to provide engineering students the tools necessary to perform calculations. This means that the authors don' really care if you know why a theorem is what it is, they just want you to be able to go out and apply already proven theorems to real life problems. It will not be until much, much later, if you decide to major in math, that you will enter into the world of whys, and finally escape the drab world of how tos.
If you have trouble with my previous post, and you would still like to know WHY the integral is sometimes interpreted as an area under the curve of the integrand, then you must venture out and resarch it. Continue to ask your questions, because this is a noble enterprise. One day, you will be a man. The road to in depth understanding of math is long and lonely. You will one of the few singularities of peculiararity that will wonder at the language of nature, whilst you marvel at the rabble, thinking to yourself, "Don't they ever wonder WHY?"
Best of luck kid.