Arc Length: Calculus Applications and Techniques

Further Applications of Calculus: Arc Length

Distance Along a Curve

The concept of arc length allows us to calculate the distance traveled along a curve, rather than simply between two points. This is a fundamental application of definite integrals in calculus, especially when the path is not a straight line.

• Key Point 1: The arc length is the actual distance along a curve between two points, not just the straight-line (Euclidean) distance.

• Key Point 2: Approximating arc length can be done by connecting points with straight segments, but calculus provides an exact formula using integration.

• Example: Consider the curve between points (1, 1) and (3, 1). The arc length is the sum of infinitesimal line segments connecting these points along the curve.

Arc Length Formula Derivation

To derive the arc length formula, we consider a small segment of the curve. Using the Pythagorean theorem, the length of a segment between (x1, y1) and (x2, y2) is:

• Key Point 1: The segment length is $\Delta s = \sqrt{(\Delta x)^2 + (\Delta y)^2}$.

• Key Point 2: By factoring and using derivatives, we obtain $\Delta s = \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$.

• Key Point 3: Summing all such segments via integration gives the arc length.

Definition of Arc Length

The arc length of a smooth curve $y = f(x)$ from $x = a$ to $x = b$ is given by:

• Formula: $s = \int_a^b \sqrt{1 + [f'(x)]^2} dx$

• Alternate Formula: For $x = g(y)$ from $y = c$ to $y = d$, $s = \int_c^d \sqrt{1 + [g'(y)]^2} dy$

• Application: Choose the formula based on which variable is easier to integrate.

Steps for Calculating Arc Length

To find the arc length of a curve, follow these steps:

1. Graph the segment: Visualize the curve and the interval.

2. Differentiate: Find $f'(x)$ or $g'(y)$ as needed.

3. Establish your integral: Set up the arc length formula.

4. Evaluate the integral: Compute the definite integral.

Example: Arc Length of $y = 2/3 x^{3/2} + 2$ from $x = 0$ to $x = 1$

Let $y' = \sqrt{x}$. The arc length is:

• Formula: $s = \int_0^1 \sqrt{1 + (\sqrt{x})^2} dx = \int_0^1 \sqrt{1 + x} dx$

• Evaluation: $s = \frac{2}{3}(2\sqrt{2} - 1)$

Example: Arc Length of $y^3 = x^2$

Sometimes, integrating with respect to $x$ is difficult. In such cases, solve for $x$ in terms of $y$ and use the alternate formula.

• Key Point: $x = \sqrt{y^3}$, $dx/dy = \frac{3}{2} y^{1/2}$

• Formula: $s = \int_0^1 \sqrt{1 + \left(\frac{dx}{dy}\right)^2} dy$

• Substitution: Use $u = 1 + \frac{9}{4} y$ for easier integration.

Example: Arc Length of a Line Segment

For a straight line between two points, the arc length is simply the Euclidean distance:

• Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Example: Between (1, 3) and (3, 7): $d = 2\sqrt{5}$

Example: Arc Length of $y = (1 - x^2)^{1/2}$ from $x = -1$ to $x = 1$

This curve is a semicircle. The arc length integral is:

• Formula: $s = \int_{-1}^1 \frac{1}{\sqrt{1 - x^2}} dx$

• Evaluation: The result is $\pi$, the length of the upper half of a unit circle.

Special Case: Catenary Curve and Area

Length of a Catenary Curve

The catenary curve, given by $y = \cosh(x)$, models freely hanging cables. Its arc length between $x = a$ and $x = b$ is:

• Formula: $\ell = \int_a^b \sqrt{1 + [\sinh(x)]^2} dx = \int_a^b \cosh(x) dx = \sinh(b) - \sinh(a)$

• Application: This formula is useful in physics and engineering for cable and chain modeling.

Area Under the Catenary Curve

Interestingly, for the catenary curve $y = \cosh(x)$, the area under the curve between $x = a$ and $x = b$ equals the arc length over the same interval:

• Formula: $A = \int_a^b \cosh(x) dx = \sinh(b) - \sinh(a) = \ell$

• Key Point: This property is unique to the catenary and one other function.

Additional info: The other function with this property is $y = e^x$, where the area under the curve from $a$ to $b$ equals the arc length over the same interval.