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Arc Length and Surface Area

Arc Length and Surface Area. Compiled by Mrs. King. Start with something easy. The length of the line segment joining points (x 0 ,y 0 ) and (x 1 ,y 1 ) is. (x 1 ,y 1 ). (x 0 ,y 0 ). www.spsu.edu/math/Dillon/2254/.../ arc hives/ arclength / arclength . ppt. The Length of a Polygonal Path?.

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Arc Length and Surface Area

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  1. Arc Length and Surface Area Compiled by Mrs. King

  2. Start with something easy The length of the line segment joining points (x0,y0) and (x1,y1) is (x1,y1) (x0,y0) www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt

  3. The Length of a Polygonal Path? Add the lengths of the line segments. www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt

  4. The length of a curve? Approximate by chopping it into polygonal pieces and adding up the lengths of the pieces www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt

  5. Approximate the curve with polygonal pieces? www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt

  6. What are we doing? • In essence, we are subdividing an arc into infinitely many line segments and calculating the sum of the lengths of these line segments. • For a demonstration, let’s visit the web.

  7. The Formula:

  8. Arc Length • Note: Many of these integrals cannot be evaluated with techniques we know. We should use a calculator to find these integrals. phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt

  9. Example Problem • Compute the arc length of the graph of over [0,1].

  10. Now comes the fun part… • First, press the Math button and select choice 9:fnInt( • Next, type the function, followed by X, the lower bound, and the upper bound. • Press Enter and you get the decimal approximation of the integral!

  11. Example • Find the arc length of the portion of the curve on the interval [0,1] phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt

  12. You try • Find the arc length of the portion of the curve on the interval [0,1] phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt

  13. Surface Area Compiled by Mrs. King

  14. Review: • Find the volume of the solid created by rotating about the x-axis on the interval [0,2] Picture from: http://math12.vln.dreamhosters.com/images/math12.vln.dreamhosters.com/2/2d/Basic_cubic_function_graph.gif

  15. Surface Area of Solids of Revolution • When we talk about the surface area of a solid of revolution, these solids only consist of what is being revolved. • For example, if the solid was a can of soup, the surface area would only include the soup can label (not the top or bottom of the can) phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt

  16. What are we doing? • Instead of calculating the volume of the rotated surface, we are now going to calculate the surface area of the solid of revolution

  17. The Formula:

  18. Ex 2.5 • Find the surface area of the surface generated by revolving about the x-axis phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt

  19. Closure • Hand in: Find the surface area of the solid created by revolving about the x-axis phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt

  20. Homework • Page • #

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