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Calculus II - 9

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Stewart Calculus Section 8.1

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Calculus II - 9

  1. 1. 8.1 Arc Lengthy 0 a b x
  2. 2. The Arc Length Formula:If is continuous on [ , ], then the lengthof the curve = ( ), , is = + [ ( )]
  3. 3. Ex: Find the perimeter of a circle with radius . o R
  4. 4. Ex: Find the perimeter of a circle with radius .Consider = , o R
  5. 5. Ex: Find the perimeter of a circle with radius .Consider = , o R = +
  6. 6. Ex: Find the perimeter of a circle with radius .Consider = , o R = + = =
  7. 7. Ex: Find the perimeter of a circle with radius .Consider = , o R = + = =Therefore the perimeter is .
  8. 8. Ex: Find the length of the arc of the hyperbola = from the point ( , ) to ( , / ).
  9. 9. Ex: Find the length of the arc of the hyperbola = from the point ( , ) to ( , / ). = , =
  10. 10. Ex: Find the length of the arc of the hyperbola = from the point ( , ) to ( , / ). = , = + = + =
  11. 11. Ex: Find the length of the arc of the hyperbola = from the point ( , ) to ( , / ). = , = + = + =Approximation method is needed, e.g. Simpson’s rule.
  12. 12. Ex: Find the length of the arc of the hyperbola = from the point ( , ) to ( , / ). = , = + = + =Approximation method is needed, e.g. Simpson’s rule. + ( )= , = , = .
  13. 13. Ex: Find the length of the arc of the hyperbola = from the point ( , ) to ( , / ). = , = + = + =Approximation method is needed, e.g. Simpson’s rule. + ( )= , = , = . . [ ( )+ ( . )+ ( . )+ ( . )+·+ ( . ) + ( )] .
  14. 14. The Arc Length Function:the length from a fixed starting point ( , ( ))to point ( , ( )) : ( )= + [ ( )]
  15. 15. The Arc Length Function:the length from a fixed starting point ( , ( ))to point ( , ( )) : ( )= + [ ( )]Ex: Find the arc length function for the curve = taking ( , ) as starting point.
  16. 16. The Arc Length Function:the length from a fixed starting point ( , ( ))to point ( , ( )) : ( )= + [ ( )]Ex: Find the arc length function for the curve = taking ( , ) as starting point. ( )= + [ ( )]
  17. 17. The Arc Length Function:the length from a fixed starting point ( , ( ))to point ( , ( )) : ( )= + [ ( )]Ex: Find the arc length function for the curve = taking ( , ) as starting point. ( )= + [ ( )] = + = +

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