2. 7.4 Lengths of Curves

3. Quick Review

4. What you’ll learn about A Sine Wave Length of a Smooth Curve Vertical Tangents, Corners, and Cusps Essential Question How can we use definite integrals to find the length of a smooth curve?

5. Example The Length of a Sine Wave 1.What is the length of the curve y = sin x from x = 0 to x = 2  ? Partition [0, 2  ] into intervals so short that the pieces of curve lying directly above the intervals are nearly straight. Each arc is nearly the same as the line segment joining its two ends. The length of the segment is: The sum over the entire partition approximates the length of the curve. Rewrite as a Riemann’s sum.

6. Example The Length of a Sine Wave Rewrite the last square root as a function evaluated at some c in the kth subinterval. Use the Mean-Value Theorem for differentiable function to obtain the sum: Take the limit as the norms of the subdivisions go to zero:

7. Arc Length: Length of a Smooth Curve If a smooth curve begins at ( a, c ) and ends at ( b, d ), a < b, c < d, then the length (arc length) of the curve is: if y is a smooth function of x on [ a, b]; if x is a smooth function of y on [ c, d].

8. Example Applying the Definition 2.Find the length of the curve y = x 2 for 0 < x < 1. This is continuous on [0, 1].

9. Example A Vertical Tangent 3.Find the length of the curve between (–1, –1) and ( 1, 1). Because the derivative is undefined at x = 0, change the equation to x as a function of y.

10. Example Getting Around a Corner 4.Find the length of the curve y = | x + 1| for –2 < x < 1. Because the derivative is undefined at x = –1, change the equation to a piecewise function.

11. Pg. 416, 7.4 #1-29 odd and #35, 37