I'm to find the acrlength parameter for s = s(t) for the path
x = e^at cos bt i + e^at sin bt j + e^at k
I do know the formula for the arclength parameter, but I'm not quite sure how the unit vectors would fit in. The formula for arclength is
s(t) = (integral from a to t)||x'(b)||db
Do I just pretend they're not there and proceed normally? Thanks!
I'm worried about the unit vectors because it's the only exercise in this chapter which has them, and there's no examples or even a correct answer given to it...
I figured I could just take the derivative normally, which would give:
x = e^at cos bt i + e^at sin bt j + e^at k
x'(t) = (ae^at cos bt - e^at sin bt) + (ae^at sin bt + e^at cos bt) + (aet)
is this correct?
Yes you're right, so the correct derivative should be:
x'(t) = (ae^at b cos bt - e^at b sin bt) + (ae^at b sin bt + e^at b cos bt) + (a e^at)
and arclength parameter:
(integral from a to t) (root)( (ae^ax b cos bx - e^ax b sin bx)^2 + (ae^ax b sin bx + e^ax b cos bx)^2 + (ae^ax)^2 )
Which isn't the easiest integral I've ever seen. I'm guessing I have to make some trig identities appear. Perhaps the first step would be:
integral from a to t) (root)( (ae^ax b (cos bx - sin bx) )^2 + (ae^ax b (sin bx + cos bx) )^2 + (ae^ax)^2 )
but then what? Thanks again!
The last one doesn't contain the b, but I could of course write:
(integral from a to t) (root)( (ae^ax b)^2 (cos bx - sin bx)^2 + (ae^ax b)^2 + (sin bx + cos bx)^2 + (ae^ax)^2 )
=
(integral from a to t) (root)( (ae^ax)^2 (b cos bx - b sin bx)^2 + (ae^ax)^2 + (b sin bx + b cos bx)^2 + (ae^ax)^2 )
=
(integral from a to t) (ae^ax)(root)((b cos bx - b sin bx)^2 + (b sin bx + b cos bx)^2)
See anything else? Thanks alot
(integral from a to t) (ae^ax)(root)((b cos bx - b sin bx)^2 + (b sin bx + b cos bx)^2)
=
(integral from a to t) (ae^ax)(root)((b cos bx)^2 - (2b cos bx sin bx) + (b sin bx)^2 + (b cos bx)^2 + (2b cos bx sin bx) + (b sin bx)^2 )
=
(integral from a to t) (ae^ax)(root)((b cos bx)^2 + (b sin bx)^2 + (b cos bx)^2 + (b sin bx)^2 )
=
(integral from a to t) (ae^ax)(root)((b^2 (cos^2 bx + sin^2 bx)^2 + b^2 (cos^2 bx + sin^2 bx)^2)
=
(integral from a to t) (ae^ax)(root)(2b^2)
=
(integral from a to t) (bae^ax)(root)2
Does this look right?