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 \(x -\text{axis} .\) Figure 6.39 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 6.39 A representative line segment over the interval \(\left[\right. y_{i - 1} , y_{i} \left]\right. .\)

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

Theorem 6.5

Arc Length for x = g(y)

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

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

Example 6.20

Calculating the Arc Length of a Function of y

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

Solution

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

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

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

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

Checkpoint 6.20

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

This lesson is part of:

Applications of Integration

View Full Tutorial

Track Your Learning Progress

Sign in to unlock unlimited practice exams, tutorial practice quizzes, personalized weak area practice, AI study assistance with Lexi, and detailed performance analytics.