8.1 Arc Lengthy    0   a                b   x
The Arc Length Formula:If   is continuous on [   , ],   then the lengthof the curve    = ( ),              ,   is         ...
Ex: Find the perimeter of a circle with radius   .                                      o          R
Ex: Find the perimeter of a circle with radius   .Consider      =           ,                                      o      ...
Ex: Find the perimeter of a circle with radius   .Consider      =             ,                                      o    ...
Ex: Find the perimeter of a circle with radius   .Consider      =             ,                                      o    ...
Ex: Find the perimeter of a circle with radius   .Consider      =             ,                                      o    ...
Ex: Find the length of the arc of the hyperbola       = from the point ( , ) to ( , / ).
Ex: Find the length of the arc of the hyperbola       = from the point ( , ) to ( , / ).        =   ,     =
Ex: Find the length of the arc of the hyperbola       = from the point ( , ) to ( , / ).        =   ,     =               ...
Ex: Find the length of the arc of the hyperbola       = from the point ( , ) to ( , / ).        =   ,     =               ...
Ex: Find the length of the arc of the hyperbola       = from the point ( , ) to ( , / ).          =   ,           =       ...
Ex: Find the length of the arc of the hyperbola       = from the point ( , ) to ( , / ).          =    ,             =    ...
The Arc Length Function:the length from a fixed starting point   ( , ( ))to point   ( , ( )) :           ( )=         + [ ...
The Arc Length Function:the length from a fixed starting point           ( , ( ))to point   ( , ( )) :           ( )=     ...
The Arc Length Function:the length from a fixed starting point             ( , ( ))to point   ( , ( )) :           ( )=   ...
The Arc Length Function:the length from a fixed starting point             ( , ( ))to point   ( , ( )) :             ( )= ...
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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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