Finding the area under the graph (integration) 
I just wanted to ask, shouldn't the value of $y$ change since the height of each rectangles are different? So why is the sum of all the strips equal to $  \sum y \delta x$? Isn't that telling us that $y$ is the same for every rectangle?
Also, how did they go from 
$$\lim_{\delta x \to 0 } \sum y \delta x$$
to
$$A = \int y \ dx$$
How did the integral symbol appear? I thought we use the integral symbol to find the equation of the function. Does that mean $ \sum y \delta x \ $ is the slope of the function?
Thank you. I'm sorry for the bad explanation 
 A: The value of $y$ does change as you say, it's implicit that $y=f(x)$ is a function of $x$. You can write $y(x)$ if it's more clear to you. And the integral symbol appeared because the limit of that sum is the definition of the Riemann integral. To be more specific, you can cut your functions in rectangles in two different ways: either like in the figure, so that the left corner of the rectangle touches the curve, or so that the right corner touches it.

In the picture, the blue rectangles are the first way, the red ones are the second way.
Say your function is defined on the interval $[a,b]$. Define $\delta x=\frac{b-a}{n}$ and $x_k=a+k\delta x$ for $k=1,2,\dots, n$, this is the k-position of the k-th rectangle. We can write the sum of the areas of the two kinds of rectangle this way. For the blue:
$$S_l(n)= \sum\limits_{k=0}^n f(x_k)\delta x$$
It's the height of the rectangle,$f(x)$, multiplied by its base $\delta x$. This is called the Riemann left sum. For the red
$$S_r(n)= \sum\limits_{k=0}^{n-1} f(x_k+\delta x)\delta x$$
It's again the height of the rectangle,$f(x+ \delta x)$, because we are taking the right corner this time, multiplied by its base $\delta x$. This is called the Riemann right sum.
You can see that as $n$ increases, both sums approach the area under the curve. But mathematicians want to be precise, so let $n$ go to infinity, if $$\lim\limits_{n\rightarrow\infty} S_r \qquad \lim\limits_{n\rightarrow\infty} S_l $$
both exist and
$$\lim\limits_{n\rightarrow\infty} S_r =\lim\limits_{n\rightarrow\infty} S_l$$
Then we say that $f$ is Riemann integrable and we write
$$ \int_a^b f(x) \mathrm{d}x=\lim\limits_{n\rightarrow\infty} S_r =\lim\limits_{n\rightarrow\infty} S_l $$
The integral sign appears because this is the definition of the integral. Some introductory courses like to present the integral as an antiderivative, but the fact that $$ \int_a^b f'(x)\mathrm{d}x=f(b)-f(a) $$
if $f$ is differentiable is a theorem, called the fundamental theorem of calculus, that can be proven starting from this definition and some other theorems about continuous and differentiable functions.
As a side note, if you are curious about just what the heck do areas have to do with slopes, I suggest you watch the youtube series "The essence of calculus" by the channel 3Blue1Brown, it's a wonderful intuitive and visual little introduction to the basic ideas of calculus and covers this and other problems.
