6.4 Arc length of a curve and surface area  (Page 2/8)

We summarize these findings in the following theorem.

Note that we are integrating an expression involving $f^{\prime }(x),$ so we need to be sure $f^{\prime }(x)$ is integrable. This is why we require $f(x)$ to be smooth. The following example shows how to apply the theorem.

Although it is nice to have a formula for calculating arc length, this particular theorem can generate expressions that are difficult to integrate. We study some techniques for integration in Introduction to Techniques of Integration . In some cases, we may have to use a computer or calculator to approximate the value of the integral.

Arc length of the curve x = g ( y )

We have just seen how to approximate the length of a curve with line segments. If we want to find the arc length of the graph of a function of $y,$ we can repeat the same process, except we partition the $y\text{-axis}$ instead of the $x\text{-axis}.$ [link] shows a representative line segment.

Then the length of the line segment is $\sqrt{{\left(\text{Δ}y\right)}^2+{\left(\text{Δ}{x}_i\right)}^2},$ which can also be written as $\text{Δ}y\sqrt{1+{\left(\left(\text{Δ}{x}_i\right)\text{/}\left(\text{Δ}y\right)\right)}^2}.$ If we now follow the same development we did earlier, we get a formula for arc length of a function $x=g(y).$