Concern regarding infinitesimal errors When finding out the area under a curve,we divide it into many thin strips and them sum up.My question is when we write for example $dW=F×ds$ and then integrate in physics terminology,we are referring to a width of $ds$ and a height of $F$.But in a curve we cannot find a perfect rectangle because the two heights won't be the same,though they will be close but not exact.If calculus is the measure of precision,why can we then write $dW=F×ds$ while we know the area of the small region will not exactly be the area of the rectangle and there will be some errors.So when we add all the infinitely thin rectangles,won't there be a small error since we are actually leaving space?I mean even when we have infinitely small width,it still isn't a perfect rectangle. P
 A: Yes that is quite a genuine doubt which arises in the mind if every beginner calculus student. No doubt the error will always be there, but we can make that error as small as WE please..... And if we make the sizes of the rectangles tending towards zero, we will get  the error tending towards zero..... Essentially this is what forms the fundamental theorem of calculus
Theoretically yes, we can get zero error as per Mathematics. And intheoretical physics too, we can tend the small areas towards zero. The infinitesimal is defined as the smallest, as small as possible. And any finite addition or subtraction does not affect it. The too small errors in the calculation of the rectangles area time to zero as does  the Dx element
In the sense of practical physics and computational sciences where we practically cannot achieve 0, we have to take Dx as per our computational capacity and conditions.
And does an error of about 0.001% really matter? The more small Dx you use the more smaller error and more the computational resources required.
