Approximate the length of the graph of \(f(x)=\sqrt{9-x^2}\) on \([−3, 3]\), using four line segments and the distance formula. Then make a better approximation with eight line segments.

Consider the given function, \(f(x)=\sqrt{9-x^2}\).

Divide the curve into four line segments on \([-3,3]\).

The graph is,

Now find the distance of each segment using the distance formula between two points.

\(\begin{align*}L_1&=\sqrt{\left(-3+2\right)^{2}+\left(f\left(-3\right)-f\left(-2\right)\right)^{2}}\\L_1&=2.4495\\L_2&=\sqrt{\left(0+2\right)^{2}+\left(f\left(0\right)-f\left(-2\right)\right)^{2}}\\L_2&=2.1409\\L_3&=\sqrt{\left(2-0\right)^{2}+\left(f\left(2\right)-f\left(0\right)\right)^{2}}\\L_3&=2.1409\\L_4&=\sqrt{\left(3-2\right)^{2}+\left(f\left(3\right)-f\left(2\right)\right)^{2}}\\L_4&=2.4495\end{align*}\)

Now sum the length of all segments.

The length of the arc approximately is \(9.1808\).

First, break the interval into 8 segments.

\([-3,-2.25],[-2.25,-1.5],[-1.5,-0.75],[-0.75,0],[0,0.75],[0.75,1.5],[1.5,2.25],[2.25,3]\)

There are 8 segments as shown below.

Using the distance formula find the lengths of all segments.

\(L=L_1+L_2+L_3+L_4+L_5+L_6+L_7+L_8\)

\(L=2.1213+0.9691+0.8102+0.7561+0.7561+0.8102+0.9691+2.1213\)

\(L\approx 9.3135\)