2.5 Arc length

Let $f(x)$ be a function. You’ve seen the formula (in MATH102) for the length along the curve $y=f(x)$ between $x=a$ and $x=b$:

But as we saw above, sometimes we are interested in parametrized curves $\gamma :ℝ\to {ℝ}^2$. In this section we will derive a formula for the length of an arc along a parametrized curve.

Recall that the derivative of a function $x:I\to ℝ$ at a number $t\in I\subseteq ℝ$, denoted $x^{\prime }(t)$, is defined as

(if this limit exists).

Suppose $\gamma (t)=(x(t),y(t))$. What is the length of the arc from $(x(t),y(t))$ to $(x(t+\delta t),y(t+\delta t))$? It’s the length of the vector $(x(t+\delta t)-x(t),y(t+\delta t)-y(t))$.

Thus the length of the arc from $(x(t),y(t))$ to $(x(t+\delta t),y(t+\delta t))$ is approximately

Dividing the interval $[a,b]$ up into segments of length $\delta t$, and taking the limit as $\delta t$ tends to zero, we obtain the following formula:

First of all, we calculate ${\gamma }^{\prime }(t)=(-2\sin t-1,\sqrt{3}\cos t)$. Now we try to calculate

The situation seems hopeless: how could we possibly integrate the square-root of such a function? But the first thing we should do is eliminate the ${\cos }^{2}t$ term. Substituting in ${\cos }^{2}t=1-{\sin }^{2}t$, we obtain

Since $\sin t\ge -1$ for any $t$, the positive root of ${(\sin t+2)}^2$ is $\sin t+2$. Thus the length of the curve is