Determine the length of the curve [Polar coordinates]

[Riazy]

Hi 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) SQRTdu = [2/3 * u* SQRT ] (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.

 

[Deleted member 4993]

Hi 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

Those are related steps.

?[sup:3smwlt3a]2[/sup:3smwlt3a] is an even function. So you can consider half the limit (0 to ? instead of -? to ?) and multiply the answer by 2.

Step5: Limits of 4 to 4+pi^2

When you substitute for the "integrand" - you would need to change the limits of integration accordingly. So if you substitute

u = ?[sup:3smwlt3a]2[/sup:3smwlt3a] + 4

then

when ? = 0 ? u = ???

and

when ? = ? ? u = ???

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

Thank you guys.