At points on a circle of radius a, line segments are drawn perpendicular to the plane of the circle, the perpendicular at each point P being of length ks, where s is the length of the arc of the circle measured counterclockwise from (a, 0) to P and k= , as shown here. Find the area of the surface formed by the perpendiculars along the arc beginning at (a, 0) and extending once around the circle.
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Imagine that you glue the initial point of the surface to the floor and then grab the upper corner and pull on it until the figure is completely stretched and contained in a 2D plane. In this new plane the x − axis is the arclength of the circle, and the height of the object at every point is a function of this, namely π 2 a 2 x . Thus to get the area under it in this coordinate system just compute ∫ 0 2 π a π 2 a 2 x d x = 2 π 2 a 2 4 π 2 a 2 = 2 4 = 2 .
The bounds of integration are 0 and 2 π a because you start measuring at 0 and the circumference of this circle (it's total arclength) is 2 π a .