# Distance between two points on a function.

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

• do you mean the distance on the curve or the line between These Points? – Dr. Sonnhard Graubner Nov 24 '17 at 16:05
• Distance on the curve – user171097 Nov 24 '17 at 16:06
• 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" – Jean Marie Nov 24 '17 at 16:06

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