[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
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.