mthsc206-fall-2010-notes-14.3

mthsc206-fall-2010-notes-14.3 - 14.3 Arc Length and...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
14.3 Arc Length and Curvature Recall: If C : x = f ( t ), y = g ( t ) for a t b be a parameterized curve, that is traversed once, with f and g continuous on [ a, b ] then the arc length of C is L = b a [ f ( t )] 2 + [ g ( t )] 2 dt. Definition 14.3.1 Now let C be a curve parameterized by x = f ( t ) , y = g ( t ) , z = h ( t ) , a t b, where r ( t ) = < f ( t ) , g ( t ) , h ( t ) > and r ( t ) is continuous on [ a, b ] . If C is traversed once, then the length of C is given by L = b a [ f ( t )] 2 + [ g ( t )] 2 + [ h ( t )] 2 dt = Example 14.3.1 Find the length of the curve C associated with the vector-valued function r ( t ) = < sin(2 t ) , cos(2 t ) , 2 t > from the point (0 , 1 , 0) to (0 , 1 , 4 π ) . Example 14.3.2 Find the length of the curve C represented by the vector-valued function r ( t ) = < sin t, cos t, t > on the interval 0 t 4 π . Question: How does the parametrization a ff ect the arc length of the curve? Definition 14.3.2 Let r ( t ) = < f ( t ) , g ( t ) , h ( t ) > , a t b , be a vector-valued function whose associated curve C is parameterized by x = f ( t ) , y = g ( t ) , z = h ( t ) , a t b, is piecewise smooth and is traversed exactly once on [ a, b ] . The arc length function
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern