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Chapter 6 | Applications of Integration
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 -axis instead of the x -axis. Figure 6.39 shows a representative line segment.
Figure 6.39 A representative line segment over the interval [ y i −1 , y i ].
Then the length of the line segment is ⎛ ⎝ Δ y ⎞ ⎠ 2 + ⎛ ⎝ Δ x i ⎞
⎠ 2 , which can also be written as Δ y 1+ ⎛ ⎝ ⎛ ⎝ Δ x i ⎞ ⎠ / ⎛ ⎝ 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 ).
Theorem 6.5: Arc Length for x = g ( y ) Let g ( y ) be a smooth function over an interval ⎡ ⎣ c , d ⎤
⎦ . Then, the arc length of the graph of g ( y ) from the point
⎛ ⎝ c , g ( c ) ⎞
⎛ ⎝ d , g ( d ) ⎞
⎠ to the point
⎠ is given by
(6.8)
Arc Length = ∫ c d
⎤ ⎦ 2 dy .
1+ ⎡
⎣ g ′( y )
Example 6.20 Calculating the Arc Length of a Function of y
Let g ( y ) =3 y 3 . Calculate the arc length of the graph of g ( y ) over the interval [1, 2].
Solution We have g ′( y ) =9 y 2 , so ⎡
⎤ ⎦ 2 =81 y 4 . Then the arc length is
⎣ g ′( y )
Arc Length = ∫ c d
2
⎤ ⎦ 2 dy = ∫
1+81 y 4 dy .
1+ ⎡
⎣ g ′( y )
1
Using a computer to approximate the value of this integral, we obtain ∫ 1 2 1+81 y 4 dy ≈21.0277.
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