7.4 A arc length

7.4 A arc length

• 2.
  An arc canbe approximated by adding up small line segments, using the distance formula 2 2 2 1 2 1( ) ( )d x x y y A rectifiable curve is one that has finite arc length. It is also continuous, and its graph is a smooth curve. By letting the points on the curve get closer and closer, you end up with more and more line segments.
• 3.
  2 2 1 n i i i sx y 2 2 2 2 1 n i i i i i y x x x Factor out and take square root, and you get 2 ix 2 1 1 ( ) n i i i i y x x Which results in the definition of arc length: Let y = f(x) represent a smooth curve on the interval [a, b]. 2 1 [ ] b a s f x dx
• 4.
  Similarly, for asmooth curve given by x = g(y), the arc length of g between c and d is 2 1 [ ] d c s f y dy
• 5.
  Examples: Find thearc length of the graph of 3 1 6 2 x y x in the interval [0.5, 2]
• 6.
  Example: Find thearc length of the graph of 3 2 1y x from x = 0 to x = 8, (Hint: solve for x, change over to y- limits