2. The Arc Length Formula:
If is continuous on [ , ], then the length
of the curve = ( ), , is
= + [ ( )]
3. Ex: Find the perimeter of a circle with radius .
o R
4. Ex: Find the perimeter of a circle with radius .
Consider
= ,
o R
5. Ex: Find the perimeter of a circle with radius .
Consider
= ,
o R
= +
6. Ex: Find the perimeter of a circle with radius .
Consider
= ,
o R
= +
= =
7. Ex: Find the perimeter of a circle with radius .
Consider
= ,
o R
= +
= =
Therefore the perimeter is .
8. Ex: Find the length of the arc of the hyperbola
= from the point ( , ) to ( , / ).
9. Ex: Find the length of the arc of the hyperbola
= from the point ( , ) to ( , / ).
= , =
10. Ex: Find the length of the arc of the hyperbola
= from the point ( , ) to ( , / ).
= , =
+
= + =
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. Ex: Find the length of the arc of the hyperbola
= from the point ( , ) to ( , / ).
= , =
+
= + =
Approximation method is needed, e.g. Simpson’s rule.
+
( )= , = , = .
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. The Arc Length Function:
the length from a fixed starting point ( , ( ))
to point ( , ( )) :
( )= + [ ( )]
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. 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. 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.
( )= + [ ( )]
= + = +