Can somebody explain why you take the derivative of a curve to find its length?

**MathIsOhSoHard** · April 15th 2012, 04:15 AM

Given a curve:

$G(t)=\binom{t}{y(t)}$

It goes between the two points:

$A=(t_1,y_1) B=(t_2,y_2)$

To find its length:

$Length=\int_{t_1}^{t_2}\sqrt{\left(\frac{d}{dt}G(t )\right)^2}\, dt$

What I understand is that you use the integral going from the values $t_1$ and $t_2$ to find the entire length between these two values. The squareroot and the exponential comes from the Pythagorean theorem to find the hypotenuse.

What I don't understand is why we take the derivative of the curve $G(t)$ .

**Prove It** · April 15th 2012, 04:27 AM

Arc length - Wikipedia, the free encyclopedia

Go to "Finding arc lengths by integrating".

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Can somebody explain why you take the derivative of a curve to find its length?

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