Curve length

Consider the curve given by the graph of a function f on an interval [a,b].

To find its length, we split the interval [a,b] into segments of length dx. The curve is now split into corresponding pieces. Since the pieces are so small, we may assume that each of them is actually a piece of a straight line.

The length of such a piece can be found using the Pythagoras rule. For that we need to know the sizes of horizontal and vertical projections. The horizontal size is dx. Since we have a straight line here, we find the vertical size by multiplying the horizontal size dx by the slope of this line, which is given by the derivative of f at x: dy = f ′(x)dx. Actually, given that y = f (x), this formula for the transformation of differential should not be surprising, we saw it before.

Summing up the lengths of all pieces we get the length of the curve:

Now we just have to make sure that the integral exists, for instance like this:

Fact.
Consider the curve given by the graph of a function f on an interval [a,b]. If f has a continuous derivative on [a,b], then the length of this curve is

Consider a parametric curve x = x(t), y = y(t) for t from [α,β].

Again, we have two options for finding the length. One is to consider the function f generated using y and the inverse to x, then we would use substitution in the above formula for the length.

Much easier is the approach via splitting of curve into little pieces. However, now we do not split into pieces determined by geometry (for instance by splitting the x-axis), but we split the time interval [α,β] into tiny pieces dt. Each such time subinterval determines a little piece of the curve. As above, because each piece is so tiny, we can imagine that they are straight lines and use the Pythagoras rule to determine their lengths. The substitution formula allows us to determine dx and dy:

To make this precise, we have to make sure that the integral exists, and also that no part of the curve is traveled several times - because then it would be also counted several times.

Fact.
Consider a parametric curve x = x(t), y = y(t) for t from [α,β] such that it intersects itself at most finitely many times. If x and y have continuous derivatives on [α,β], then the length of this curve is equal to

Consider a curve in polar coordinates given by ϱ = ϱ(φ) for φ from [α,β].

Again, there are two approaches possible. One is to split the angle into tiny sub-angles of size dφ. We may imagine that the corresponding pieces of the curve are actually straight lines and calculate its length using the Pythagoras rule:

Note that the "across the angle" part was calculated using the formula "radius times angle" for the arc length, because for such a short segment, the arc length and the length of a straight segment are about the same. The perpendicular part dϱ can be calculated using the substitution formula from the equality ϱ = ϱ(φ). Summing up one gets the length of the curve:

It is also possible to pass from the polar coordinates to rectangular coordinates: x = ϱ(φ)cos(φ), y = ϱ(φ)sin(φ).

This defines a parametric curve, therefore we may use the above formula:

To avoid trouble, we should make sure that no part of the curve is counted several times. We will do it the easy way:

Fact.
Consider a curve in polar coordinates given by ϱ = ϱ(φ) for φ from [α,β] which intersects itself at most finitely many times. If ϱ(φ) has a continuous derivative on [α,β], then the length of this curve is

Volume of a solid of revolution
Back to Theory - Applications