Philosophical question about the integral expression

[Jacobim]

How is it possible...that the integral performs an infinite amount of calculations to give the area under a curve.

The integral expression has to find an infinite amount of areas of the (dx by f(x)) rectangles.

I'm guessing there is a simple answer to this, I'm just not quite piecing it together.

I have probably done several homework assignments covering exactly what I'm asking, but I do not know how to answer this question to myself.

 

[Jacobim]

"We take the limit as dx approaches zero"

So this limit means there is no need to calculate the area of the infinite amount of rectangles. Is that it? I was hoping for something more exciting!

 

[HallsofIvy]

Well, its a little more complicated that just "dx approaches 0" because the number of terms is increasing at the same time, but yes, we do NOT actually calculate and infinite number of things.

 

[Curious3141]

It's like asking how you can sum an infinite series a + ar + ar² + ar³ + ... where |r|< 1 to get exactly a/(1-r) without needing an infinite number of calculations.

 

[Dick]

"We take the limit as dx approaches zero"

So this limit means there is no need to calculate the area of the infinite amount of rectangles. Is that it? I was hoping for something more exciting!

No, you have it. If you can compute limits without summing an infinite number of things then you can get the answer without doing an infinite amount of work. It's like Zeno's argument that Achilles can't overtake the tortoise. But only vaguely.