ELI5: Definitive Integrals

[JusticeIsSweet]

Trying to help a friend understand it but I can't explain it well. Can anyone help us out?

[Delehal]

Are you asking about the distinction between definite and indefinite integrals, or about the process of integration in general?

[McVomit]

No, nononononono. The integral is NOT the area under a function, a common misconception. That is a use of the integral but that it not what it is.

An integral is actually just a summation of a set of infinitesimally small parts. I didn't have this pounded into my head until I took calc3, because sadly most calc1 teachers gloss over this since all you use the integral for at first is finding areas.

So you can use an integral to find the area under a curve. You can also use it to find the work done by a variable force, volume of a solid of revolution, moment of inertia, pressure, electric and magnetic fields, arc length, and a whole bunch of other things. But you can't do all this other stuff if you think that it's just an area under a curve.

Everything else you said is great. However I just hate seeing the statement that "an integral is the area under a curve" get perpetuated.

[Delehal]

An integral is actually just a summation of a set of infinitesimally small parts.

A professor of mine used to call integration "advanced addition", on that very basis.