13.3 - Arc Length

Consider the following equation.

$$y=f(x)$$

Do you remember the Arc Length formula?

$$l={\int }_a^b\sqrt{1+(f^{\prime }(x){)}^2}dx$$

This works normally for what we have been using it for. However, How can we apply this formula to vector arithmetic?

We can use the following for 2d

$$l={\int }_{t0}^{t1}\sqrt{(\frac{dy}{dt}{)}^2+(\frac{dx}{dt}{)}^2}dt$$

We can further this by applying it in 3D.

$$l={\int }_{t0}^{t1}\sqrt{(\frac{dz}{dt}{)}^2+(\frac{dy}{dt}{)}^2+(\frac{dx}{dt}{)}^2}dt$$

We can use this in terms of r prime using the following

$${\int }_{t0}^{t1}|\overline{r^{\prime }}(t)|dt$$

Arc Length Function

We define the arc length function as s(t), where it is an integral from A to T instead

It is similar to the arc length formula, but returns a function for scalar operations

Formula:

$$s(t)={\int }_a^t|\overline{r^{\prime }}(t)|dt$$

We can differentiate it resulting in the following

$$\frac{ds}{dt}=|\overline{r^{\prime }}(t)|$$ $$\frac{4}{5}=$$