arc length and curvature. Awesome. For this assignment. You submit answers by qu

Question

arc length and curvature. Awesome. For this assignment. You submit answers by questions parts. The number of submissions remaining for each question part only changes if you submit or change the answer. Your last submission is used for your score. Find the length of the curve. r(t) = (2sin(t), 9t,2 cos (t) ,-6 t 6 L= Need Help? Read it Master it Chat About is Find the length of the curve. r(t) = 3ti + 8t3/2j+12t2k, 0 t 1 L= Need Help? Read it Chat About it Reparametrize the curve with respect to arc length measured from the where t = o in the direction of increasing t. (Enter your anser in terms of s. ) r(t) =3ti +(8-4t) j +2(+2t)k Need Help? Road it Watch it Chat About it

Explanation / Answer

1) r(t)=xi+yj+zk

Length of the curve =integral( sqrt((dx/dt)^2 +(dy/dt)^2+(dz/dt)^2)dt)

r(t) = 2sint i + 9t j + 2cost k

dx/dt = d(2sint)/dt = 2cost

dy/dt = 9

dz/dt = -2sint

(dx/dt)^2+(dy/dt)^2+(dz/dt)^2 = 4cos^2 t + 81+ 4 sin^2 t

= 4(cos^2 t+sin^2 t) +81 = 4+81 = 85

length of the curve = integral(-6 to 6){ sqrt(81) dt}

= sqrt(85) t (from -6 to 6)

=sqrt(85){ 6-(-6)}

= 12 sqrt(85)

2) r(t) = 3t i+8t^(3/2) j + 12t^2 k

dx/dt = 3

dy/dt = 8*3/2 * t^(3/2 -1) = 12t^(1/2)

dz/dt = 12*2t = 24t

perform the integration as shown in the above solution

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