Length of a Plane Curve

1. Length of a Plane Curve Objective: To find the length of a plane curve.

2. Arc Length • Our first objective is to define what we mean by length (or arc length) of a plane curve y = f(x) over an interval [a, b]. Once that is done we will be able to focus on the problem of computing arc lengths. We will have the requirement that f / be continuous on [a, b] and we say that y = f(x) is a smooth curve.

3. Arc Length Problem • Suppose that y = f(x) is a smooth curve on the interval [a, b]. Define and find a formula for the arc length of L of the curve y = f(x) over the interval [a, b]

4. Arc Length • To define the arc length of a curve we start by breaking the curve into small segments. Then we approximate the curve segments by line segments and add the lengths of the line segments to form a Riemann Sum. As we increase the number of segments, the approximation becomes better and better.

5. Arc Length • To implement our idea, divide the interval [a, b] into n subintervals by inserting points between the values . Let be the points on the curve that join the line segments. These line segments form polygonal path that we can regard as an approximation to the curve y = f(x).

6. Arc Length • The length Lk of the kth line segment in the polygonal path is

7. Arc Length • The length Lk of the kth line segment in the polygonal path is • If we now add the lengths of these line segments, we obtain the following approximation to the length L of the curve.

8. Arc Length • To put this in the form of a Riemann Sum we will apply the Mean-Value Theorem. This Theorem implies that there is a point between and such that or

9. Arc Length • To put this in the form of a Riemann Sum we will apply the Mean-Value Theorem. This Theorem implies that there is a point between and such that or

10. Arc Length • To put this in the form of a Riemann Sum we will apply the Mean-Value Theorem. This Theorem implies that there is a point between and such that or

11. Arc Length • Thus, taking the limit as n increases and the widths of the subintervals approximate zero yields the following integral that defines the arc length L:

12. Definition • 7.4.2 If y = f(x) is a smooth curve on the interval [a, b], then the arc length of L of this curve over [a, b] is defined as

13. Definition • 7.4.2 If x = g(y) is a smooth curve on the interval [c, d], then the arc length of L of this curve over [c, d] is defined as

14. Example 1 • Find the arc length of the curve from (1, 1) to (2, ) using both formulas.

15. Example 1 • Find the arc length of the curve from (1, 1) to (2, ) using both formulas.

16. Example 1 • Find the arc length of the curve from (1, 1) to (2, ) using both formulas.

17. Homework • Page 469 • 3, 5

18. Average Value of a Function • We will use the idea of average/mean and extend the concept so that we can compute not only the arithmetic average of finitely many function values but an average of all values of f(x) as x varies over a closed interval [a, b].

19. Average Value of a Function • We will use the Mean-Value Theorem for Integrals, which states that if f is continuous on the interval [a, b], then there is at least one point in this interval such that

20. Average Value of a Function • We will use the Mean-Value Theorem for Integrals, which states that if f is continuous on the interval [a, b], then there is at least one point in this interval such that • We will look at the equation in this form as our candidate for the average value of f over the interval [a, b].

21. Average Value of a Function • To explain what motivates this idea, divide the interval [a, b] into n subintervals of equal length and choosing arbitrary points in successive subintervals. Then the arithmetic average of the values is

22. Average Value of a Function • Using the fact that • we can say that • and substitute to get this equation:

23. Average Value of a Function • Taking the limit as yields

24. Average Value of a Function • Definition 7.6.1 If f is continuous on [a, b], then the average value (or mean value) of f on [a, b] is defined to be

25. Average Value of a Function • If we look at the Mean-Value Theorem for Integrals together with the equation for average value, we can see the relationship.

26. Average Value of a Function • The Mean-Value Theorem for Integrals guarantees the point where the rectangle has the right height. The average value is the height.

27. Example 1 • Find the average value of the function over the interval [1, 4], and find all points in the interval at which the value of f is the same as the average.

28. Example 1 • Find the average value of the function over the interval [1, 4], and find all points in the interval at which the value of f is the same as the average. • The average value of the functions is:

29. Example 1 • Find the average value of the function over the interval [1, 4], and find all points in the interval at which the value of f is the same as the average. • The second question is the Mean-Value Theorem for Integrals.

30. Average Velocity Revisited • When we first looked at Rectilinear Motion, we defined the average velocity of the particle over a time interval to be its displacement over the time interval divided by the time elapsed. Thus, if the particle has position s(t), then its average velocity over a time interval [t0 , t1 ] is

31. Average Velocity Revisited • However, the displacement is the integral of velocity over the given time interval. We can now look at average velocity as:

32. Homework • Pages 479-480 • 1-9 odd • Section 5.8 • Pages 388-389 • 1-11 odd