Question

. Find the arc length of the curve r(t) = <t^2 cos(t), t^2 sin(t)> from the point (0, 0) to (−π^2 , 0).

Answer #1

1. a) Get the arc length of the curve.
r(t)= (cos(t) + tsin(t), sin(t) - tcos(t), √3/2 t^2) in
the Interval 1 ≤ t ≤ 4
b) Get the curvature of r(t) = (e^2t sen(t), e^2t, e^2t cos
(t))

Find an arc length parametrization of
r(t) =
(et
sin(t),
et
cos(t),
10et )

Calculate the arc length of the indicated portion of the curve
r(t).
r(t) = 6√2 t^((3⁄2) )i + (9t sin t)j + (9t cos t)k ; -3 ≤ t ≤
7

Find, for 0 ≤ x ≤ π, the arc-length of
the segment of the curve R(t) = ( 2cost − cos2t, 2sint −
sin2t )
corresponding to 0 ≤ t ≤ x.

Consider the curve r(t) = cost(t)i + sin(t)j +
(2/3)t2/3k
Find:
a. the length of the curve from t = 0 to t = 2pi.
b. the equation of the tangent line at the point t = 0.
c. the speed of the point moving along the curve at the point t
= 2pi

With the parametric equation x=cos(t)+tsin(t), y=sin(t)-tcos(t)
, 0 ≤ t ≤ 2π)
Find the length of the given curve. (10 point)
2) In the circle of r = 6, the area
above the r = 3 cos (θ) line
Write the integral or integrals expressing the area of this
region by drawing. (10 point)

Find the arc length of the curve r(t) = i + 3t2j +
t3k on the interval [0,√45].
Hint: Use u-substitution to integrate.

Write the curve given by r(t)=((3/2)t)i+(t^3/2)j as a function
r(s) parameterized by the arc length s from the point where t=0.
Write your answer using standard unit vector notation.

1) Find the length of the parametric curve x=2 cos(t)
, y=2 sin(t) on the interval [0, pi].
2) A rope lying on the floor is 10 meters long and its
mass is 80 kg. How much work is required to raise one end of the
rope to a height of 15 meters?

Parametrize the curve r(t) = <5sin(t), 5cos(t), 12t> with
respect to arc length, measured from the point (0,5,0).

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