For the complete revolution (0<=t<=2pi)  of the helix curve x=cost  and y=sint and z=t evaluate integration (y.sinz ds)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the arclength of the given curve, in three dimensions, such that:

`int_0^(2pi) sqrt((x'(t))^2 + (y'(t))^2 + (z'(t))^2) dt`

You need to differentiate x,t and z with respect to t, such that:

`x'(t) = (cos t)' => x'(t) = -sin t`

`y'(t) = (sin t)' => y'(t) = cos t`

 

`z'(t) = t'=> z'(t) = 1`

`int_0^(2pi) sqrt((-sin t)^2 + (cos t)^2 + 1^2) dt`

`int_0^(2pi) sqrt(sin^2 t + cos^2 t + 1) dt`

Using the fundamental formula of trigonmetry yields:

`sin^2 t + cos^2 t = 1`

`int_0^(2pi) sqrt(1 + 1) dt => int_0^(2pi) sqrt2 dt`

`int_0^(2pi) sqrt2 dt = sqrt2*t|_0^(2pi)`

Using the fundamental formula of calculus yields:

`int_0^(2pi) sqrt2 dt = sqrt 2(2pi - 0) = 2sqrt2*pi`

Hence, evaluating the arclength of the given curve, in three dimensions, yields `int_0^(2pi) sqrt2 dt = 2sqrt2*pi` .

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