# How do you find the distance between two points on a parabola

So I've been wanting to figure out a formula for an odd pattern I found... but to write a proof, I need to know one thing...

How do I find the distance between two points on a parabola?

Like, if I have y=x^2, how do I find out how much actual line is between say, (4,16) and (2,4)? If you use the normal line distance thing, it tells you how much space is between each point, but not how much of the actual parabola is between the points.

So can you help me out?

Thanks.

• Do you want the length of the curve between the two points? – user17762 May 12 '13 at 7:06
• planetmath.org/arclengthofparabola – lab bhattacharjee May 12 '13 at 7:06
• @iostream007 Regarding your suggested edit. The OP is asking for the length of the curve between the two points and not the area between the two points. So I would suggest that you do not repeatedly edit it as area. – user17762 May 12 '13 at 7:25

If you have a curve $y=f(x)$, the length of the curve between points $(x_1,f(x_1))$ and $(x_2,f(x_2))$ is given by $$L(x_1,x_2) = \int_{x_1}^{x_2} \sqrt{(dx)^2 + (df(x))^2} = \int_{x_1}^{x_2}\sqrt{1+\left(\dfrac{df}{dx} \right)^2} dx$$ In your case, $f(x) = x^2$. Hence, $\dfrac{df}{dx} = 2x$. Hence, the length of curve between the points $(x_1,x_1^2)$ and $(x_2,x_2^2)$ is $$L(x_1,x_2) = \int_{x_1}^{x_2} \sqrt{1+(2x)^2} dx = \int_{x_1}^{x_2} \sqrt{1+4x^2} dx$$ Can you evaluate this integral?

$$\int_{x_1}^{x_2} \sqrt{1+4x^2} dx = \left.\dfrac{2x \sqrt{4x^2+1} + \log(2x+\sqrt{1+4x^2})}4 \right \vert_{x_1}^{x_2}$$

EDIT

Added picture to highlight the different lengths. • This is also a really great answer, just one question. Because you include f(x)= x^2 in the equation, do you no longer need the y coord? Thanks. – Lucas May 12 '13 at 16:30
• Actually scratch that I understand it, you don't need the y coord in this, got it. One last thing though, if I plugged in 4 and 2 to that crazy formula, then subtracted to find the length, would I get 12? (4^2-2^2=12). I'm just guessing, because if it doesn't, I've made a mistake in my thinking. – Lucas May 12 '13 at 16:38
• @Lucas You wont and should not get $4^2 - 2^2 = 12$. $12$ is the difference between the $y$ coordinates, which is different form the length of the curve. I have added a figure to clarify what the different lengths are. – user17762 May 12 '13 at 18:04

Rectification (that's what this is called) of the curve given by $f(x)$ is obtained by the integral $$\int_a^b \sqrt{1+f'(x)^2}\,\mathrm dx.$$ Hre you need to evaluate $$\int_2^4\sqrt{1+4x^2}\,\mathrm dx.$$