SECTION 7 4 ARC LENGTH ARC LENGTH v

SECTION 7. 4 ARC LENGTH

ARC LENGTH v. What do we mean by the length of a curve? v. We might think of fitting a piece of string to the curve here and then measuring the string against a ruler. 7. 4 P 2

ARC LENGTH v. However, that might be difficult to do with much accuracy if we have a complicated curve. v. We need a precise definition for the length of an arc of a curve—in the same spirit as the definitions we developed for the concepts of area and volume. 7. 4 P 3

POLYGON v. If the curve is a polygon, we can easily find its length. n n We just add the lengths of the line segments that form the polygon. We can use the distance formula to find the distance between the endpoints of each segment. 7. 4 P 4

ARC LENGTH v. We are going to define the length of a general curve in the following way. n n First, we approximate it by a polygon. Then, we take a limit as the number of segments of the polygon is increased. 7. 4 P 5

ARC LENGTH v. This process is familiar for the case of a circle, where the circumference is the limit of lengths of inscribed polygons. 7. 4 P 6

ARC LENGTH v. Now, suppose that a curve C is defined by the equation y = f(x), where f is continuous and a ≤ x ≤ b. v. We obtain a polygonal approximation to C by dividing the interval [a, b] into n subintervals with endpoints x 0, x 1, …, xn and equal width Dx. 7. 4 P 7

ARC LENGTH v. If yi = f(xi), then the point Pi (xi, yi) lies on C and the polygon with vertices Po, P 1, …, Pn, is an approximation to C. 7. 4 P 8

ARC LENGTH v. The length L of C is approximately the length of this polygon and the approximation gets better as we let n increase, as Figure 4. 7. 4 P 9 Fig. 9. 1. 3, p. 561

ARC LENGTH v. Here, the arc of the curve between Pi– 1 and Pi has been magnified and approximations with successively smaller values of Dx are shown. 7. 4 P 10

Definition 1 v. Thus, we define the length L of the curve C with equation y = f(x), a ≤ x ≤ b, as the limit of the lengths of these inscribed polygons (if the limit exists): 7. 4 P 11

ARC LENGTH v. Notice that the procedure for defining arc length is very similar to the procedure we used for defining area and volume. n n n First, we divided the curve into a large number of small parts. Then, we found the approximate lengths of the small parts and added them. Finally, we took the limit as n → ∞. 7. 4 P 12

ARC LENGTH v. The definition of arc length given by Equation 1 is not very convenient for computational purposes. n However, we can derive an integral formula for L in the case where f has a continuous derivative. v. Such a function f is called smooth because a small change in x produces a small change in f ’(x). 7. 4 P 13

SMOOTH FUNCTION v. If we let Dyi = yi – yi– 1, then 7. 4 P 14

SMOOTH FUNCTION v. By applying the Mean Value Theorem to f on the interval [xi– 1, xi], we find that there is a number xi* between xi– 1 and xi such that is, 7. 4 P 15

SMOOTH FUNCTION v. Thus, we have: 7. 4 P 16

SMOOTH FUNCTION v. Therefore, by Definition 1, 7. 4 P 17

SMOOTH FUNCTION v. We recognize this expression as being equal to vby the definition of a definite integral. n This integral exists because the function is continuous. v. Therefore, we have proved the following theorem. 7. 4 P 18

THE ARC LENGTH FORMULA If f ’ is continuous on [a, b], then the length of the curve y = f(x), a ≤ x ≤ b is: v. If we use Leibniz notation for derivatives, we can write the arc length formula as: 7. 4 P 19

Example 1 v. Find the length of the arc of the semicubical parabola y 2 = x 3 between the points (1, 1) and (4, 8). (See Figure 5. ) 7. 4 P 20

Example 1 SOLUTION v. For the top half of the curve, we have: v. Thus, the arc length formula gives: 7. 4 P 21

Example 1 SOLUTION v. If we substitute u = 1 + (9/4)x, then du = (9/4) dx. v. When x = 1, u = 13/4. v. When x = 4, u = 10. v. Therefore, 7. 4 P 22

ARC LENGTH v. If a curve has the equation x = g(y), c ≤ y ≤ d, and g’(y) is continuous, then by interchanging the roles of x and y in Formula 2 or Equation 3, we obtain its length as: 7. 4 P 23

Example 2 v. Find the length of the arc of the parabola y 2 = x from (0, 0) to (1, 1). v. SOLUTION n n Since x = y 2, we have dx/dy = 2 y. Then, Formula 4 gives: 7. 4 P 24