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  3. suppose c is some curve in the ry plane the...

Question: suppose c is some curve in the ry plane the...

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Suppose C is some curve in the ry plane. The ribbon R of witdh 2h around C is the set of points in the plane whose distance from C is at most h. Below you see a typical ribbon example where C (the black curve) is a half circle, and R is bounded by the red curves. For a general curve C, to parametrize D, the best way is to start with the arclength parametrization of C Note that in this case. P(s)l-i and-the length of C (a) Sketch a figure and explain why the vector n(sg (s) i f (s)j. is the unit normal vector to the curve C at the point r (s) set of points P x = f(s)+tg(s), (b) Sketch a figure and explain why the ribbon R can be seen as the T(s) tn(s): namely y = g(s)-tf(s), 0 s L, -h t h c) Use the change of variables ry to st and compute the area of the ribbon. Your answer should be 2hL (d) The area of the half ribbon 0 th is much more complicated as you will see from the integral in part (d). Can you give a geometric explanation bu roughly sketching a curve and the corresponding half ribbon? (e) Using geometry, check your result in case of the half circle C of radius 1. Note that the half ribbon above C is larger than the half ribbon below, but the sum becomes 2Lh (f) Explain in the case of the circle that this computation will not make sense if h >1

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