We're asked to find a function s(t), for the arc length of a curve centered at point t=0, as a function of t. The function is as follows...
$\gamma (t) =e^t i + \sqrt{2} tj-e^{-t}k$
My work is as follows...
$\gamma '(t) =e^t i + \sqrt{2} j+e^{-t}k$
$s(t)=\int_{-t}^{+t} \sqrt{(e^x)^2+(\sqrt{2})^2+(e^{-x})^2} \,dx$
$s(t)=\int_{-t}^{+t} \sqrt{e^{2x}+2+e^{-2x}} \,dx$
...Used computer to compute integral(side question: what method could be used to integrate by hand)...
$s(t)=e^x-e^{-x} \Big|_{-t}^{+t}$
$s(t)=e^t-e^{-t}-(e^{-t}-e^t)$
Final Answer: $s(t)=2e^t-2e^{-t}$
However, my answer is marked as incorrect. I don't believe there's something wrong with the calculating bits, cause if I graph my answer vs the computer calculating the whole thing, I get the same line- Picture Of Graph. So it leads me to believe I'm doing something wrong in the setup. Pretty sure the arc length equation is fine. So the only thing I can think of is I'm interpreting the "centered at point t=0" incorrectly?
Thanks in advance for the help!
Yes, I suspect they intended to say “starting at $t=0$” and you interpreted their “centered at” literally. (Is this a question written in English or a translation?)
To answer your other question, you should recognize the perfect square under the square root. You will probably encounter a number of them.