Question

- 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)

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))

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.

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

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.

Consider the parametric equations below.
x = t sin(t), y = t
cos(t), 0 ≤ t ≤ π/3
Set up an integral that represents the area of the surface
obtained by rotating the given curve about the x-axis.
Use your calculator to find the surface area correct to four
decimal places

7. For the parametric curve x(t) = 2 − 5 cos(t), y(t) = 1 + 3
sin(t), t ∈ [0, 2π) Part a: (2 points) Give an equation relating x
and y that represents the curve. Part b: (4 points) Find the slope
of the tangent line to the curve when t = π 6 . Part c: (4 points)
State the points (x, y) where the tangent line is horizontal

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?

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

2. Rotate the semicircle of radius 2 given by y = √(4 − x^2)
about the x-axis to generate a sphere of radius 2, and use this to
calculate the surface area of the sphere.
3. Consider the curve given by parametric equations x = 2
sin(t), y = 2 cos(t).
a. Find dy/dx
b. Find the arclength of the curve for 0 ≤ θ ≤ 2π.
4.
a. Sketch one loop of the curve r = sin(2θ) and find...

Question(9) Curve ? = t - sin t, y = 1 - cos t, 0 ≤ ? ≤ 2? given.
a) Take the derivatives of x and y according to t and arrange them.
b) Write and edit the integral that gives the surface area of the object formed by rotating the given curve around the x axis.
c) Solve the integral and find the surface area.

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