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Length of Curve Calculator

f(x) =
Left Endpoint (a) =
Right Endpoint (b) =
Solution:

Length of Curve Lesson

Lesson Contents

What is the Length of a Curve?

The length of a curve, which is also called the arc length of a function, is the total distance traveled by a point when it follows the graph of a function along an interval [a, b].

To visualize what the length of a curve looks like, we can pretend a function such as y = f(x) = x2 is a rope that was laid down on the x-y coordinate plane starting at x = -2 and ending at x = 2.

This rope is not pulled tight since it is laid down in the shape of a parabola. However, if we were to pull on both ends it would become a tight, linear rope. The length of this rope line is the length of the curve y = f(x) = x2 from x = -2 to x = 2, notated as the interval [-2, 2].

How to Calculate the Length of a Curve

The formula for calculating the length of a curve is given as:

$$\begin{align} L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \: dx \end{align}$$

Where L is the length of the function y = f(x) on the x interval [a, b] and is the derivative of the function y = f(x) with respect to x.

The arc length formula is derived from the methodology of approximating the length of a curve. To approximate, we break a curve into many segments. If each segment is treated as a straight line, we may use the distance formula to determine each line’s length.

Adding up the lengths from these many straight lines gives an approximation of the curve’s length. The accuracy of that approximation gets better as we break the curve into a greater number of shorter straight lines.

After setting up the distance formula for the length of these line segments we may use an integral to make those line segments infinite in quantity and infinitesimally short. Each small change in x value is the dx from the arc length formula. In fact, the arc length formula is a simplified summation of an infinite number of distance formula evaluations for the straight lines.

Example Problem 1

$$\begin{align}
& \text{1.) Find the length of } y = f(x) = x^{2} \text{ between } -2 \leq x \leq 2 \hspace{20ex} \\ \\ & \hspace{3ex} \text{ Using the arc length formula } \: L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \: dx \\ \\ & \text{2.) Given } y = f(x) = x^2\text{, find } \frac{dy}{dx} : \\ \\ & \hspace{3ex} \frac{dy}{dx} = 2 \cdot x\\ \\ & \text{3.) Plug lower x limit } a \text{, upper x limit } b \text{, and } \frac{dy}{dx} \text{ into the arc length formula} : \\ \\ & \hspace{3ex} L = \int_{-2}^{2} \sqrt{1 + \left(2 \cdot x\right)^2} \: dx\\ \\ & \text{4.) Solve for length by evaluating the integral}: \\ \\ & \hspace{3ex} L = \int_{-2}^{2} \sqrt{1 + \left(2 \cdot x\right)^2} \: dx = 9.2936 \end{align}$$

Example Problem 2

$$\begin{align}
& \text{1.) Find the length of } y = f(x) = \ln \left(x\right) \: – \: 2x \text{ between } \frac{2}{3} \leq x \leq 12 \hspace{20ex} \\ \\ & \hspace{3ex} \text{ Using the arc length formula } \: L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \: dx
\\ \\ & \text{2.) Given } y = f(x) = \ln \left(x\right) \: – \: 2x \text{, find } \frac{dy}{dx} : \\ \\ & \hspace{3ex} \frac{dy}{dx} = – 2 + \frac{1}{x}\\ \\ & \text{3.) Plug lower x limit } a \text{, upper x limit } b \text{, and } \frac{dy}{dx} \text{ into the arc length formula} : \\ \\ & \hspace{3ex} L = \int_{\frac{2}{3}}^{12} \sqrt{1 + \left(- 2 + \frac{1}{x}\right)^2} \: dx\\ \\ & \text{4.) Solve for length by evaluating the integral}: \\ \\ & \hspace{3ex} L = \int_{\frac{2}{3}}^{12} \sqrt{1 + \left(- 2 + \frac{1}{x}\right)^2} \: dx = 22.8515 \end{align}$$

How the Calculator Works

The Voovers Length of Curve Calculator is written in the web programming languages HTML (HyperText Markup Language), CSS (Cascading Style Sheets), and JS (JavaScript). The HTML creates the architecture of the calculator, the CSS provides the visual styling, and the JS provides all functionality and interactiveness.

When the “calculate” button is clicked, your inputted function and interval are read by the JS routine. The routine calls on a JS computer algebra system (CAS) which can perform a derivative symbolically, maneuvering equations and applying derivative rules just like a person!

The CAS performs the differentiation to find dydx. Then, that expression is plugged into the arc length formula. The integral is evaluated, and that answer is rounded to the fourth decimal place. The final length of curve result is printed to the answer area along with the solution steps.

Then, a large list of (x, y) coordinate pairs is generated for the function. These coordinate pairs are fed to a JS graphing utility that draws a smooth curve through the points. This function plot is displayed below the solution steps.

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