The arc length, \( s \) along the graph of a parametric function \( (x,y) = \left(x(t),y(t)\right) \) from \( t=a \) to \( t=b \) is given by:
We can see why by examining a linear approximation of the arclength:
by the Pythagorean Theorem, which means that for very small , .
Also, notice that in the the case where , we have , and the arc length becomes:
which we can use for non-parametrized functions.
Show that the arc length of a semi-circle of radius 1 is .
Let , which describes the unit-circle. Then the arc length around the the semi-circle is given by:
But by the Pythagorean Identity, so
Application and Extensions
Consider the surface area and volume of of the solid formed by rotating the region bounded by the -axis, , and , around the -axis. This solid is called Gabriel's Horn.
The volume of the solid of revolution can be found using the disk method:
Now consider what happens as we allow to approach infinity:
The surface area of a solid of revolution is given by the formula
In this case, since that gives us:
This integral is hard to evaluate, but since in our interval and :
It follows that:
But , which means that Gabriel's Horn has infinite surface area, but a volume of only !