3. Let C be parametrized by x=2cos3t and y=2sin3t for 0 t /2. i. Set up the inte

ID: 2856943 • Letter: 3

Question

3. Let C be parametrized by x=2cos3t and y=2sin3t for 0 t /2.
i. Set up the integral for the length L of C. DO NOT evaluate the integral.

ii. (Extra credit: 3 points) Evaluate the integral in (i) and obtain the length L of C. Let C be parametrized by x=1+2sin3t and y=12cos3t, for0 t 4.

(7) Show that C is a circle. Identify the center and radius, and indicate on the graph of C the initial point P(0) .

(8) Indicate the direction of increasing t (with reason), and determine the number of revolutions traversed as t increases from 0 to 4.

x = 2cos3t , y=2sin3t for 0 t /2

dx/dt = 2[ -sin(3t) (3)] = -6sin(3t)

dy/dt = 2[ cos(3t) (3)] = 6cos(3t)

Length of arc L = ds

ds = [(dx/dt)2 + (dy/dt)2] dt

==> L = [0 to /2] [(-6sin(3t))2 + (6cos(3t))2] dt

ii) L = [0 to /2] [(-6sin(3t))2 + (6cos(3t))2] dt

==> [0 to /2] [36sin2(3t) + 36cos2(3t)] dt

==> [0 to /2] [36(sin2(3t) + cos2(3t))] dt

==> [0 to /2] 6[(sin2(3t) + cos2(3t)] dt

==> [0 to /2] 6[1] dt since sin2x + cos2x = 1 ; here x = 3t

==> [0 to /2] 6dt

==> 6 [0 to /2] (t)

==> 6 [ (/2) - (0) ]

==> 3

Hence length of curve L = 3 units = 9.425 units

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