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I want to find distance between two points lying on a function $f(x)=x^2$ and points are $(2,4)$ and $(3,9)$

Could someone please show me how this is done.. by solving the above question and also please mention the general formula for it. Thank you

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  • $\begingroup$ do you mean the distance on the curve or the line between These Points? $\endgroup$ – Dr. Sonnhard Graubner Nov 24 '17 at 16:05
  • $\begingroup$ Distance on the curve $\endgroup$ – user171097 Nov 24 '17 at 16:06
  • $\begingroup$ You mean distance between 2 points on a $\color{red}{curve}$ ? The fact that they are on a curve doesn't change the formula you have surely seen for the distance between "ordinary points" $\endgroup$ – Jean Marie Nov 24 '17 at 16:06
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If you want to find the straight line distance then you just have to use the Pythagorean Theorem to get:

$$d = \sqrt{(3-2)^2 + (9-4)^2} = \sqrt{1 + 25} = \sqrt{26}$$

If you want to calculate the length along the curve $y=x^2$ then you'll need to calculate the arclength. This can be done by the following formula:

$$d = \int_2^3 \sqrt{1 + (y')^2}dx = \int_2^3 \sqrt{1 + 4x^2} dx$$

The following integral can be evaluated by using the tan substitution for $x$.

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