Arc Length Proof
How would you calculate arc length for a curve? Before choosing an option, think about how each of the following possibilities might or might not yield a solution.
A . Create a bunch of Riemann rectangles, like with Riemann sums, and add up the lengths of the uppermost line segments, creating a Riemann staircase. Take the limit of all such totals.
B . Flatten out the curve by rotating each part of it to become horizontal. Then, take the length of that line.
C . Approximate the curve with a lot of line segments, and take the sum of their length. Take the limit of all such sums.
D . Draw the curve with a 1-unit think pen, so that a vertical 1-unit line segment bisected by the function is contained in the region. The length of the curve is then equal to the area of that region, which can be found with traditional calculus methods.
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1 solution
The only answer that is entirely correct is C.
Method A fails to give the correct answer because "staircases" do not accurately measure hypotenuse length (try it for any right triangle). A popular proof that π = 4 falls prey to this realization.
Method B is nearly correct, but an infinite number of rotations would be needed to "flatten" any curve that is actually curvy. However, there is no guarantee that an (uncountably) infinite number of rotations would maintain arc length-- parts of the curve may collapse back onto itself, for instance.
Method D fails to give the correct answer because the line segments are vertical and not necessarily orthogonal to the function. The described method provides an accurate estimate for the x -distance traversed by the function.