Question

Find the length of the curve given by
r(t)=sqrt(2)/2t i+e^(t/2) j+e^(-t/2) z

where -1<=t<=5.

Answer 1

http://www.algebra.com/services/rendering/simplifier.mpl?expression=r%2A%28t%29=s%2Aqr%2At%2A%282%29%2F2%2Ati%2Be^%28t%2F2%29%2Aj%2Be^%28-t%2F2%29%2Az
I think you should use this website

Answer 2

r'(t) = ( sqrt(2) / 2 , (1/2) e^(t/2) , (-1/2)e^(-t/2) )

Answer 3

\[| r'(t)| = \sqrt{ (1/2) + (1/4)e^{t} +(1/4)e^{-t} }= \sqrt{ e^{-t} [ (1/2)e^{t} +(1/4)e^{2t} + (1/4) ]}\]

Answer 4

\[|r'(t)|= \sqrt{ \frac{e^{-t}}{2} [ \frac{1}{2} (e^{t})^2 + e^{t} +\frac{1}{2}] } = \sqrt{ \frac{e^{-t}}{2} \times \frac{1}{2} (e^{t}+1)^2}\]

Answer 5

\[|r'(t)| = \frac{1}{2} e^{- \frac{t}{2}} (e^{t} +1) = \frac{1}{2} ( e^{\frac{t}{2} } + e^{- \frac{t}{2}} )\]

Answer 6

Therefore answer is simple

Answer 7

\[= \frac{1}{2} \int\limits_{1}^{5} ( e^{\frac{t}{2}} + e^{- \frac{t}{2}} ) dt\]