Arc Length of Curves: Calculus Study Notes

Length of Curves (Arc Length)

Introduction to Arc Length

The arc length of a curve describes the distance along the curve between two points. In calculus, this is computed using integrals, allowing for precise measurement of curves defined by functions.

• Arc length is the total distance along a curve between two points.

• For a function y = f(x), the arc length from x = a to x = b is found using calculus.

Approximating Arc Length

Arc length can be approximated by dividing the curve into small segments and summing their lengths. As the number of segments increases, the approximation becomes more accurate.

• First approximation: Use the distance formula between two points: $\sqrt{(b-a)^2 + (f(b)-f(a))^2}$

• Second approximation: Divide the interval into two segments and sum their lengths.

• Nth approximation: Divide the interval into n segments, each of width $\Delta x = \frac{b-a}{n}$, and sum the segment lengths.

Arc Length Formula

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

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

• This formula is derived from the Pythagorean theorem applied to infinitesimal segments of the curve.

Example: Arc Length of a Straight Line

For a straight line y = mx + b, the arc length between two points is equivalent to the distance formula.

• Derivative: $f'(x) = m$

• Arc length: $L = \int_{x_0}^{x_1} \sqrt{1 + m^2} \, dx = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$

• This confirms the arc length formula generalizes the distance formula for straight lines.

Example: Arc Length of $y = x^{3/2}$ from $x = 0$ to $x = 4$

To find the arc length of the curve $y = x^{3/2}$ between $x = 0$ and $x = 4$, follow these steps:

• Derivative: $f'(x) = \frac{3}{2} x^{1/2}$

• Arc length formula: $L = \int_0^4 \sqrt{1 + \left(\frac{3}{2} x^{1/2}\right)^2} \, dx = \int_0^4 \sqrt{1 + \frac{9}{4} x} \, dx$

• Substitute $u = 1 + \frac{9}{4}x$, $du = \frac{9}{4}dx$ to simplify the integral.

• Result: $L = \frac{8}{27} \left[(10)^{3/2} - 1\right]$

Example: The graph below illustrates the curve $y = x^{3/2}$ and the interval of interest.

Example: Arc Length of $y = 2e^x + e^{-x}$ on $[0, \ln 2]$

For the curve $y = 2e^x + e^{-x}$, the arc length is calculated as follows:

• Derivative: $f'(x) = 2e^x - e^{-x}$

• Arc length formula: $L = \int_0^{\ln 2} \sqrt{1 + (2e^x - e^{-x})^2} \, dx$

• Expand and simplify the integrand, then integrate over the interval.

• Result: $L = \frac{33}{16}$

Example: Circumference of a Circle Using Arc Length Formula

The circumference of a circle of radius r can be derived using the arc length formula. Consider the upper half of the circle $x^2 + y^2 = r^2$.

• Express y: $y = \sqrt{r^2 - x^2}$

• Derivative: $f'(x) = \frac{-x}{\sqrt{r^2 - x^2}}$

• Arc length formula: $L = 2 \int_{-r}^{r} \sqrt{1 + \left(\frac{-x}{\sqrt{r^2 - x^2}}\right)^2} \, dx$

• Simplify to $L = 2 \int_{-r}^{r} \frac{r}{\sqrt{r^2 - x^2}} \, dx$

• Use trigonometric substitution: $x = r \sin \theta$, $dx = r \cos \theta \, d\theta$

• Bounds: $x = -r \rightarrow \theta = -\frac{\pi}{2}$, $x = r \rightarrow \theta = \frac{\pi}{2}$

• Result: $L = 2\pi r$

Example: The diagram below shows the trigonometric substitution and the sine function relevant to the calculation.

Summary Table: Arc Length Formulas

Curve	Arc Length Formula	Example Result
Straight Line	$L = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$	Distance between two points
General Function $y = f(x)$	$L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx$	See examples above
Circle of radius $r$	$L = 2\pi r$	Circumference