# Thread: The length of a curve with polar coordinates

1. ## The length of a curve with polar coordinates

[Edited below] see the third post for a more detailed attempt to solve it

Hi guys,

Problem: A curve is given in polar coordinates by r = fi^2 between -pi <= fi =< pi

Determine the length of the curve.

My progress:

I recall a formula my teacher has given us: L(x) = $(using given limits of integration) SQRT (h^2(fi)) + [h'(fi)]^2) d(fi) = This would give me$SQRT (fi^2)^2 + (2fi)^2 d(fi)

So far so good, right?

$SQRT (fi)^4 + 4fi^2 d(fi) I think Variable substititution would be good here U = (fi)^4 + 4fi^2 d(fi) DU/D(fi) = 4(fi)^3 + 8fi Hmm How can i complete this? could someone help me in detail, thanks 2. Originally Posted by Riazy Hi guys, Problem: A curve is given in polar coordinates by r = fi^2 between -pi <= fi =< pi Determine the length of the curve. My progress: I recall a formula my teacher has given us: L(x) =$(using given limits of integration) SQRT (h^2(fi)) + [h'(fi)]^2) d(fi)

= This would give me

$SQRT (fi^2)^2 + (2fi)^2 d(fi) So far so good, right?$SQRT (fi)^4 + 4fi^2 d(fi)

I think Variable substititution would be good here

U = (fi)^4 + 4fi^2 d(fi)

DU/D(fi) = 4(fi)^3 + 8fi

Hmm How can i complete this?
could someone help me in detail, thanks

If I am reading this correctly you have

$\displaystyle r=\phi^2$

The arc length formula in polar coordinates is

$\displaystyle \displaystyle s=\int_{-\pi}^{\pi}\sqrt{r^2+\left(\frac{dr}{d\phi} \right)^2}d\phi$

Plugging in gives

$\displaystyle \displaystyle \int_{-\pi}^{\pi}\sqrt{(\phi^2)^2+(2\phi)^2}d\phi=\int_{a }^{b}\phi\sqrt{\phi^2+4}d\phi$

3. Hi again guys I would like if someone here could explain the steps in the following problem:

Problem:

A curve is given by polar coordinates by r= (fi)^2, between -pi <= fi <= pi
Determine the length of the curve.

Solution ( I have got it from a friend, but don't understand all of it)
// "I am not able to get in touch with him//

1. We know that r = f(fi) = fi^2, and that its between -pi <= fi <= pi.

2. We can write this as r^2 + ('r)^2 = fi^4 +(2fi)^2 = fi^4 + 4fi^2 = fi^2(fi^2+4)

3. ds = SQRT [r^2+(r')^2] dfi = I(Fi)I (<--- Absolute value of fi, by taking SQRT) ->
I(Fi)I SQRT[fi^2 +f] dfi -> S = $(limits of integration from - pi to pi)I(Fi)I SQRT[fi^2+4] dfi 4 = 2$ (from 0 to pi) SQRT[fi^2 + 4fi] dfi = /u= fi^2 + 4 =-> du = 2fi(dfi) / =

5. \$(limits of integration from 4 to 4+pi^2) SQRT[u]du = [2/3 * u* SQRT [u] ] (from 4 to pi + 4)

= 2/3 (( pi^2 + 4)^3/2 -8)

The thing is that I find many of the steps confusing and I don't understand what has really been done

for example
Step 4: Where does the constant 2 come from?, how did the limits of integration change into 0-pi
Step5: Limits of 4 to 4+pi^2

I would like if someone could make those to steps in a little more detail,

Thank you guys.