Definition of integral calculus?

Integral calculus is absolutely nothing more than finding the total quantity of something by firstly splitting it up into many equally sized units, then adding those units up to find the overall quantity.
Do you agree with that?

For example, if you know that the answer to the below problem is "six" , then you fully understand essentially what integral calculus actually is about....

Tommy eats...
1 mars bar on Monday,
1 mars bar on tuesday,
1 mars bar on wednesday,
1 mars bar on thursday,
2 mars bar on friday,
no mars bars on Saturday
no mars bars on sunday

How many mars bars does tommy eat in a week?

The definition you wrote is that of "series": the discrete sum of things.
The integral is similar but llittle bit more complex since it is the sum of infinitesimal quantities. The simbol \[\int\] comes from an ancient "S" that stands for Sum
We have two cases:

1. Indefinite integral
2. Definite integral between "a" and "b"

The first doesn't lead to a number but to the function called antiderivative
The second is instead a number. Calling F(.) the antiderivative it is given by F(b)-F(a); it also has a geometrical meaning: the area of the curve under integration, calculated between "a" and "b".

thanks, I did indeed describe discrete integration, but its pretty much the same thing.
In electronics, I am sure you would agree that we are mostly only interested in Definite integrals between a and b.
Eg integrating a periodic current waveform in order to find the RMS.

In any case, do you agree that anybody who understands my mars bar example essentially understands integral calculus. in electronics we are not interested in indefinite integrals, at least not for general switch mode power supply design etc.

Its just like understanding that a car that travelled at 60mph for one hour has travelled 60 miles in one hour.........if anyone worked that out then they have essentially done integration in their head..do you agree?

Not really.
Much control theory relies on second order ODEs where the appropriate limits for a definite solution are sometimes not obvious before you have the indefinite form.
This stuff matters when optiising the feedback compensation networks in your power supplies.

Then we have things like the information preserving transforms (Many of which are integrals over the range minus infinity to infinity, think Fourier, Laplace, things of that nature, poles and zeros are kind of important to filter design).

Maxwells EM wave equations are similar, and that is before you consider things like line and surface integrals (Useful for figuring out field strengths and charge distributions).

What you describe are (at best) numerical approximations (Think trapezium rule) which are sort of useful sometimes, but having an algebraic solution usually allows for things like finding poles and zeros by examination which you cannot do with a numerical result.

Regards, Dan.

"The area under the curve" they said :/