Using Technology to Find the Arc Length of an Ellipse

Use technology to find the arc length \(s\) of the ellipse shown in Figure 17 traced out by the vector function \(\mathbf{r}( t) =2\cos t\mathbf{i}+3\sin t\mathbf{j}\) from \(t=0\) to \(t=\dfrac{\pi }{2}.\)

Solution We begin by finding \(\mathbf{r^{\prime} }( t) \) and \(\left\Vert \mathbf{r^{\prime} }( t) \right\Vert .\) \[ \mathbf{r}^{\prime} (t)=-2\sin t\mathbf{i}+3\cos t\mathbf{j}\qquad \left\Vert \mathbf{r}^{\prime} (t)\right\Vert =\sqrt{4\sin ^{2}t+9\cos ^{2}t} \]

Now we use the formula for arc length. \begin{equation*} s=\int_{a}^{b}\left\Vert \mathbf{r}^{\prime} (t)\right\Vert dt=\int_{0}^{\pi /2}\sqrt{4\sin ^{2}t+9\cos ^{2}t}\,dt \end{equation*}

This is an integral that has no antiderivative in terms of elementary functions. To obtain a numerical approximation to the arc length, we use a CAS or a graphing utility.

When entering the integral in WolframAlpha, we find \begin{equation*} \int_{0}^{\pi /2}\sqrt{4\sin ^{2}t+9\cos ^{2}t}\,dt\approx 3.96636 \end{equation*}

If we use a TI-84 to find the value of this integral, we obtain the same result. The screen shot is given in Figure 18.