Reborn in Fire (hammerforge) wrote in calculus,
Reborn in Fire
hammerforge
calculus

Determining arc length of a vector.

So, I am in Calculus 3 right now and we are covering vector calculus. This week we are covering arc lengths.
While I understand the formula for arc length, I have a specific problem that I do not understand the answer which the book computed. It is not so much a question of the application of the formula as something the book did in its solutions manual without any obvious reason.
The problem is formated as:

r(t) = 21/2t i + etj + e-t k

r'(t) = 21/2 + et -e-t


|r'(t)| = ( (21/2)2 + (et)2 + (-e-t)2)(1/2)
This is where my answer diverges from the books...According to my understanding of math, 21/221/2=21/2. According to the book, the '2' simply disapears.

After arbitrarilly dropping the '2', the book continues on its merry way. with:

|r'(t)| = (e2t + e-2t)1/2

|r'(t)| = ((et + e-t)2)1/2

|r'(t)| = (et + e-t)

Integrate |r'(t)| ... and so on and so forth...

What I fail to understand is how and why the '2' is arbitrarilly dropped. In my understanding, the 2 should stay in and be integrated to a value of 21/2t . Resultantly my answer and the books answer differ by the value of 21/2.

Can anyone explain what happens to the 2?
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