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SOLVED: “Integrals” & “integrator”: clear … AS MUD
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TimFox:
My favorite example of a useful limit to a non-obvious question:
Consider the expression sin(x)/x.  At x=0, this becomes 0/0, which is "indeterminate".
However, in the limit as x->0, the limit of the ratio becomes 1.
You can verify this by inspection, with an old-fashioned trigonometry table (for angles in radians), or with a calculator (again, using radian angles).
Or, you can use the theorem that states that the limit of a ratio equals the ratio of the limits of the derivatives of numerator and denominator.
RJSV:
   Here is an example to try; take a simple drive, in your car, and write down your speed.  Do very boring same speed and result, you can have a simple diagram, a rectangle.
That rectangle is just speed, or rate multiplied by time, giving your miles total.  Now, do a couple others, separate and simple, so you've then gotten FOUR separate and different sizes, separate little runs in car.
Add those up, (and don't get distracted by talk of area under curve...too abstract).
   Now then, see the trend,...it's to go smaller and finer, each time taking thinner 'slices' in the rectangular pieces.  Well, obtaining integrals, or integration, is the goal of smoothness, of the process outcome, in other words, if you go way down, to zero thick (pieces).
See diagram of that process of going finer on the horiz. scale.
RJSV:
   The tricky process, of figuring out the first integrals, I learned that derivatives, more easily understood intuitively, sit in a heirarchy, along with integrals, where integrals are 'more complex' or higher up the ladder so to speak.
   The first integrals were easy, and math folks, (I heard, lol), math folks learned to pick things apart by going backwards, through the process, and came up with a thoerm or two.  I'm thinking the SINE COSINE periodic functions, with obvious slope of 'one' at zero cross locations, was an easy integral.
There was always a constant came out, unknown, that also needed to be solved / defined.
   So you know your area, but need to solve for 'the limit', as the pieces approach 'zero' thick, and having the function shape, vertically.  I think the approach also had to use 'trapazoid' shaped pieces, sloped top as it varies in height from point to point.  That's one false trap to avoid...each shrunken piece does not assume zero slope top, it just as sloped as bigger sized.
TimFox:
Only a slight exaggeration, and why integration scares people:  https://xkcd.com/2117/
rstofer:
If you want to use slices to integrate, you are headed toward Riemann Sums.  There are 3 simple sums before we get to the trapezoidal approach.  They are the left, center and right sum where the name is which corner or center of the rectangle is being used to create the slice.  The center sum is the best approximation of the three.  The trapezoidal considers the trapezoid at the top of each slice and is also a good approximation.

https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/a/left-and-right-riemann-sums

Here are the results of integrating f(x) = (x**2) + sqrt(1.0 + (2.0 * x)) between 3 and 5 using 100,000 slices:


--- Code: ---Sums taken over   100000 slices
Left   Riemann Sum =  38.65404
Center Riemann Sum =  38.65420
Right  Riemann Sum =  38.65437

Trapezoidal Area   =  38.65420

Definite Integral  =  38.65420...

--- End code ---

The center and trapezoidal integrals are quite close to the calculated definite integral.



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