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

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

Arc Lengths of Curves Cont.

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 $x\text{-axis}.$

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 $\left[{y}_{i-1},{y}_{i}\right].$

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

arc length for $x$ = $g$ ( $y$ )

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

Arc Length = \int_{c}^{d} \sqrt{1 + {[g^{'} (y)]}^{2}} d y

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

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

We have ${g}^{\prime }(y)=9{y}^{2},$ so ${\left[{g}^{\prime }(y)\right]}^{2}=81{y}^{4}.$ Then the arc length is

Arc Length = \int_{c}^{d} \sqrt{1 + {[g^{'} (y)]}^{2}} d y = \int_{1}^{2} \sqrt{1 + 81 y^{4}} d y .

$\text{Arc Length}={\displaystyle\int }_{c}^{d}\sqrt{1+{\left[{g}^{\prime }(y)\right]}^{2}}dy={\displaystyle\int }_{1}^{2}\sqrt{1+81{y}^{4}}dy.$

Using a computer to approximate the value of this integral, we obtain

\int_{1}^{2} \sqrt{1 + 81 y^{4}} d y \approx 21.0277.

${\displaystyle\int }_{1}^{2}\sqrt{1+81{y}^{4}}dy\approx 21.0277.$

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

$\text{Arc Length}=3.15018$

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

Arc Lengths of Curves Cont.

Arc Length of the Curve xxx = ggg(yyy)

arc length for xxx = ggg(yyy)

Arc Length of the Curve $x$ = $g$ ( $y$ )

arc length for $x$ = $g$ ( $y$ )