The document discusses the arc length formula and how to calculate arc length. The arc length formula is the integral of the square root of (dx/dt)^2 + (dy/dt)^2 from t1 to t2, where t1 and t2 are the starting and ending points. Examples are provided on calculating the perimeter of a circle using this formula and approximating the arc length of a hyperbola between two points.

1. 8.1 Arc Length
y
0 a b x

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.
( )= + [ ( )]
= + = +