Find the length of the curve given by r (t) = ((2^0.5)/2) t)i+(e^t/2)j+(e^−t/2)k, where −2 =< t =< 7.

okay - so let me take a shot at it. Since you are given a function r(t) in terms of all vector component, what needs to be done is to take partial derivatives of each components in terms of t. Then square each terms and add them, put the sum of squared partials under square root and integrate over the range of t then that should be the length of your curve.

i did that and got ((1/2)+(1/4)e^t+(1/4)e^-t )^(1/2) so i have to integrate that.

that would be something like... \[s = \int\limits_{t_{i}}^{t_{f}} \sqrt((\delta x/\delta t)^2 + (\delta y/\delta t)^2 + (\delta z/\delta t)^2)) dt\]

something along this line

be sure that dt is not under square roots

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