Question

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))

Answer #1

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)

r(t)=[cos(t),sin(t),cos(3t)]
r(t)=[tcos(t),tsin(t),t)
r(t)=[cos(t),sin(t),t2]
r(t)=[t2cos(t),t2sin(t),t]
r(t)=[cos(t),t,sin(t)]
Sketch the graphs.

Sketch the graph of each curve by finding surfaces on which they
lie.
(A) <t,tcos(t),tsin(t)>
(B) <cos(t),sin(t),sin(2t)>
(C) <t,cos(t),sin(t)>
Please Sketch in a 3D graph.

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

1. Graph the curve given in parametric form by x = e t sin(t)
and y = e t cos(t) on the interval 0 ≤ t ≤ π2.
2. Find the length of the curve in the previous problem.
3. In the polar curve defined by r = 1 − sin(θ) find the points
where the tangent line is vertical.

Find the length of the curve
1) x=2sin t+2t, y=2cos t, 0≤t≤pi
2) x=6 cos t, y=6 sin t, 0≤t≤pi
3) x=7sin t- 7t cos t, y=7cos t+ 7 t sin t, 0≤t≤pi/4

Consider the parametric curve given by the equations:
x = tsin(t) and y = t cos(t) for 0 ≤ t ≤ 1
(a) Find the slope of a tangent line to this curve when t =
1.
(b) Find the arclength of this curve

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

Consider the following vector function.
r(t) =
6t2, sin(t) − t cos(t), cos(t) + t sin(t)
, t > 0
(a) Find the unit tangent and unit normal vectors
T(t) and
N(t).
T(t)
=
N(t)
=
(b) Use this formula to find the curvature.
κ(t) =

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