How do you find the length of a curve in calculus?
In Cartesian coordinates for y = f(x) defined on interval
In general, we could just write:
Let's use Cartesian coordinates for this explanation.
If we consider an arbitrary curve defined as
Consider a point on the curve
This means that the approximate total length of curve is simply a sum of all of these line segments:
If we want the exact length of the curve, then we can make the assumption that all of the points are infinitesimally separated. We now take the limit of our sum as
Since we are working in the
We can now apply the Mean Value Theorem, which states there exists a point
which we could also write (using the notation we are using) as
Applying this means we now have
Simplifying this expression a bit gives us
We can now use this new distance definition for our points in our summation.
Sums are nice, but integrals are nicer for continuous circumstances! It's easy to just write this as a definite integral since both integrals and sums are "summation" tools. In the integral, we can drop our sum index as well.
Writing this a little bit more typically yields
We have arrived at our result! In general, the length is usually defined for a differential of arclength
In general, you need to take the derivative of the function defining your curve to substitute into the integral. Then the trick is to find a way (usually) to try and get a perfect square inside the square root to simplify the integral and find your solution. It varies for every type of curve.
Let me know if you have any further questions in the comments!