29. If you unwind thread from a stationary circular spool, keeping the thread taut at all times, then the endpoint traces a curve C called the involute of the circle (Figure 9). Observe that PQ has length RO. Show that C is parametrized by c(0) = (R(cos 0+0 sin 0), R(sin 0 – 0 cos 0) Then find the length of the involute for 0 <0 < 2n. P=(x. y) FIGURE 9 Involute of a circle.

Q: Give a rough sketch of the graph of the curve r= 1+ cos40, 0 < 0<2r.

A: The given curve is  r=1+cos 4θ,  0≤θ≤2π

Q: What quadrant does the spiral r=-theta enter first as theta increases from 0 ?

A: The equation you provided, r = -θ, describes a spiral with a negative slope. In polar coordinates, θ…

Q: Find a°y*dx where C is the arc of the curve x = = y? from (0,0) to (9,3)

A:

Q: (a) Evaluate sin 2z dz, 3z -71 where C is an anticlockwise unit circle centred at the origin. 4.

A:

Q: Find the arc length function for the curve y=18x*2 with starting point P.(1, 18). O z =…

A: