1. Mrs. Cartledge Arc length
Calculus
Phantom’s Revenge Kennywood Park
Mrs. Cartledge
Arc length
Sec 7.4

2. Objectives
Determine the length of a curve.

3. Area Under a Curve (revisited)
We use the definite integral to find the area under a curve. The limits of integration are a and b. The height of the rectangle is represented by f(x), the width by dx and the sum by the definite integral.

4. Sum of Disks as the Volume (revisited)

5. Finding Arc Length
We want to determine the length of the continuous function f(x)  on the interval [a,b] .  Initially we’ll need to estimate the length of the curve.  We’ll do this by dividing the interval up into n equal subintervals each of width x  and we’ll denote the point on the curve at each point by Pi.  We can approximate the curve by a series of straight lines connecting the points. 

6. The length of one segmentP5 P4

7. The length of all segments

8. Definition of Arc Length

9. Example Applying the Definition

10. Example
Find the length of the specified curve. =

11. Algorithm to Find Arc Length
Determine whether the length is with respect to x or y, and then find the endpoints for the interval.
Find y’(x) or x’(y).
Plug the derivative into the formula:
Evaluate.

12. Sample
Find the length of the specified curve.
[1,2]

13. Sample
Find the length of the specified curve. You are given the x values but you need c and d!!

14. Sample
Find the length of the specified curve.
y = sin(x) [0, ¶ ]

15. Sample
Find the length of the specified curve.
y = (x2 – 4)2 [0, 4 ]

16. Example A Vertical Tangent

17. Sample
y = x1/5 [-1,4]

18. Closure
Explain the difference in these two formulas.

19. Independent Assignment
Notebook: p 485 # 3 - #11 odd. Check your answers in the back of the book.
Graded Assignment: HW Sec 7.4 in Schoology. Enter the first ½ of the answers in Schoology. Due Tuesday, May 15, before class.