Arc Length of a Curve and Surface Area: Learn It 2

Arc Lengths of Curves Cont.

Arc Length of the Curve [latex]x[/latex] = [latex]g[/latex]([latex]y[/latex])

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 [latex]y,[/latex] we can repeat the same process, except we partition the [latex]y\text{-axis}[/latex] instead of the [latex]x\text{-axis}.[/latex]

The figure below shows a representative line segment.

This figure is a graph. It is a curve to the right of the y-axis beginning at the point g(ysubi-1). The curve ends in the first quadrant at the point g(ysubi). Between the two points on the curve is a line segment. A right triangle is formed with this line segment as the hypotenuse, a horizontal segment with length delta x, and a vertical line segment with length delta y.
Figure 3. A representative line segment over the interval [latex]\left[{y}_{i-1},{y}_{i}\right].[/latex]

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

arc length for [latex]x[/latex] = [latex]g[/latex]([latex]y[/latex])

Let [latex]g(y)[/latex] be a smooth function over an interval [latex]\left[c,d\right].[/latex] Then, the arc length of the graph of [latex]g(y)[/latex] from the point [latex](c,g(c))[/latex] to the point [latex](d,g(d))[/latex] is given by:

[latex]\text{Arc Length}={\displaystyle\int }_{c}^{d}\sqrt{1+{\left[{g}^{\prime }(y)\right]}^{2}}dy[/latex]

Let [latex]g(y)=3{y}^{3}.[/latex] Calculate the arc length of the graph of [latex]g(y)[/latex] over the interval [latex]\left[1,2\right].[/latex]

Let [latex]g(y)=\frac{1}{y}.[/latex] Calculate the arc length of the graph of [latex]g(y)[/latex] over the interval [latex]\left[1,4\right].[/latex] Use a computer or calculator to approximate the value of the integral.