Introduction to Arc Length Now that we’re done with techniques of integration, we’ll return to doing some geometry; this will lead to some of the tools you’ll need in multivariable calculus. Our ﬁrst topic is arc length, which is calculated using another cumulative sum which will have an associated story and picture. Suppose you have a roadway with mileage markers s0 , s1 , s2 , ..., sn along the road. The distance traveled along the road — the arc length — is described by this parameter s. If we look at the road as a graph, we can let a be the x coordinate of the ﬁrst point s0 on the curve or road and b be the x coordinate of the end point sn of the curve, and xi as the x-coordinate of si . This is reminiscent of what we did with Riemann sums. ds dy ds dx Figure 1: Straight line approximation of arc length. We’ll approximate the length s of the curve by summing the straight line distances between the points si . As n increases and the distance between the si decreases, the straight line distance from si to si−1 will get closer and closer to the distance Δs along the�curve. We can use the Pythagorean theorem to see that that distance equals (Δx)2 + (Δy)2 . In other words: 2 (hypotenuse) � �� 2 (Δs) ≈ (Δx)2 + (Δy)2 . We apply the tools of calculus to this estimate; in the inﬁnitesimal this is exactly correct: (ds)2 = (dx)2 + (dy)2 . In the future we’ll omit the parentheses and write this as ds2 = dx2 +dy 2 . These are squares of diﬀerentials; try not to mistake them for diﬀerentials of squares. The next thing we do is take the square root: � ds = dx2 + dy 2 . This is the formula that Professor Jerison has memorized, but you can rewrite it in several useful ways. For instance, you can factor out the dx to get: � � �2 ds dy = 1+ . dx dx This is the form we’ll be using today; when we add up all the inﬁnitesimal values of ds we’ll ﬁnd that: Arc Length = = = distance along the curve from s0 to sn � � �2 � b dy 1+ dx dx a � ds � = b � 1 + f � (x)2 dx (y = f (x)) a Question: Is f � (x)2 equal to f �� (x)? Answer: No. Suppose f (x) = x2 . Then f � (x) = 2x, f � (x)2 = 4x2 and f �� (x) = 2. � Question: What are the limits of integration on ds, above? Answer: If you’re integrating with respect to s you’ll start at s0 and end at sn . If you’re integrating with respect to a diﬀerent variable you’ll have diﬀerent limits of integration, as happens when we change variables. The values s0 and sn are mileage markers along the road; they’re not the same as a and b. When we measure arc length, remember that we’re measuring distance along a curved path.