While finding the length of a curve, we assume an infinitesimal right triangle, of width $dx$ and height $dy$, so arc length is ${\sqrt{dx^2 + dy^2}}$. But my question is that actually the curve is not having such a triangle the curve is continuously changing according to function, not linearly. So, there will be an error in the length of a small part of the arc. Though this error may be neglected for small part but when we integrate over the whole curve the error will be finite. So, why don't we encounter the error that arises due to this part? Please explain in easy language I am just a beginner in calculus.
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$\begingroup$You can approximate the continuous curve with consecutive chords drawn from point to pint on it. The length of these chords will be given by pythagoreous formula. As you progressively draw smaller and smaller chords, the sum of the length of these chords tends to a number which is taken as the length of the curve.
$\endgroup$ $\begingroup$I think it's best to not think of $dx$ to be infinitesimal small, but instead to be just small but finite. Then there will be some error, as you correctly observed. Now, what happens if you instead choose a $dx$ which is only half as big? Well, intuitively, the error gets smaller. Now, choose a $dx$ which is again half as big, and then again and again. The error gets smaller and smaller and, in the limit, approaches zero. And this limit is how you define the length of the curve, no infinitesimals involved.
There are of course some simplifications in the description above: when making $dx$ smaller, the error not necessarily becomes smaller. However, it is sufficient if the error "eventually" becomes "sufficiently smaller" if you only continue to decrease $dx$. Also, of course, you want that the error approaches zero independently if you split $dx$ into two, three or $\pi$ parts every step...
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