Name: measuring distance along a curve
Uploaded: 2026-01-12T12:24:11.000Z
Duration: 1 min 22 s
Channel: JoVE

La función de longitud de arco muestra la distancia total recorrida a lo largo de una curva suave desde un punto de partida fijo hasta un extremo variable.Para una curva continua y diferenciable, esto se encuentra sumando pequeños segmentos lineales a lo largo de la curva. Estos segmentos aproximan la curva usando cambios horizontales y verticales, similar a una suma de Riemann.Cuando el tamaño del segmento se acerca a cero, la suma se convierte en una integral que da la longitud exacta del arco.Para expresar la longitud del arco como función, se utiliza una variable ficticia dentro de la integral, permitiendo que el límite superior varía.El integrando contiene la raíz cuadrada de uno más el cuadrado de la derivada. Siempre es mayor o igual a uno y aumenta a medida que la curva se vuelve más pronunciada, lo que hace que la longitud del arco crezca más rápido.Utilizando el Teorema Fundamental del Cálculo para diferenciar la función se obtiene la tasa de cambio de la longitud del arco, que depende directamente de la pendiente de la curva.Por ejemplo, al instalar vallas de barrera a lo largo de una carretera sinuosa, la función de longitud de arco mide con precisión la distancia al suelo, ayudando a evitar la subestimación de materiales, costes y tiempo de instalación.

función de longitud de arco

Arc Length Function

La funci&#xF3;n de longitud de arco representa la distancia total recorrida a lo largo de una curva suave, medida desde un punto inicial fijo hasta un punto final variable. Para curvas continuas y diferenciables, la longitud de arco proporciona una forma precisa de cuantificar la distancia cuando las aproximaciones con l&#xED;neas rectas son insuficientes.Para obtener la longitud de arco, la curva se divide en muchos segmentos peque&#xF1;os. Cada segmento se aproxima mediante una l&#xED;nea recta cuya longitud depende de los incrementos horizontales y verticales en ese intervalo. Estos segmentos lineales se asemejan a la estructura de una suma de Riemann. A medida que aumenta el n&#xFA;mero de segmentos y su anchura disminuye hacia cero, la aproximaci&#xF3;n converge a una integral que da la longitud exacta de la curva.Para una funci&#xF3;n y = f(x) diferenciable en un intervalo, la longitud de arco desde un punto fijo x = a hasta un punto final variable x viene dada por:\begin{equation*}L(x) = \int_a^x \bm{\sqrt{1 + (f&#x27;(u))^2}}\, du\end{equation*}El integrando siempre es mayor o igual a uno, lo que refleja que la distancia m&#xE1;s corta entre dos puntos es una l&#xED;nea recta. A medida que aumenta la magnitud de la derivada, lo que indica una curva m&#xE1;s pronunciada, el valor del integrando aumenta, lo que provoca que la longitud de arco se incremente m&#xE1;s r&#xE1;pidamente.La diferenciaci&#xF3;n de la funci&#xF3;n de longitud de arco mediante el Teorema Fundamental del C&#xE1;lculo muestra que su tasa de cambio en cualquier punto depende directamente de la pendiente de la curva en ese punto. Esto resalta la estrecha relaci&#xF3;n entre el comportamiento geom&#xE9;trico local y la distancia total acumulada.Las funciones de longitud de arco son fundamentales en aplicaciones pr&#xE1;cticas donde se requiere una medici&#xF3;n precisa de la distancia a lo largo de trayectorias curvas. Por ejemplo, al instalar vallas de barrera en una carretera con curvas, los c&#xE1;lculos de la longitud de arco garantizan que se mida la distancia real sobre el terreno, lo que ayuda a evitar la subestimaci&#xF3;n de los materiales, los costes y el tiempo de instalaci&#xF3;n.

Función de longitud de arco

La funci&#xF3;n de longitud de arco representa la distancia total recorrida a lo largo de una curva suave, medida desde un punto inicial fijo hasta un ...

Education

Educación

JoVE Core

Calculus

Cálculo

Unit

Unidad

Application of Techniques of Integration

Aplicación de Técnicas de Integración

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

measuring distance along a curve

The arc length function measures the distance traveled along a smooth curve from a fixed starting point to a moving endpoint. It is useful when a straight-line estimate is not accurate enough. For curves that are continuous and differentiable, arc length gives a precise way to track length along the curve.

To find this distance, the curve is broken into many small pieces. Each piece is treated like a short straight line. Its length depends on the horizontal change and the vertical change over that interval. These pieces work like a Riemann sum, where many small estimates are added together.

As the number of segments increases and each width gets smaller and smaller, the approximation becomes an exact integral. For a differentiable function y = f(x) on an interval, the arc length from x = a to a variable endpoint x is given by an integral. The integrand is always at least 1, because the shortest path between two points is a straight line.

The slope of the curve affects how fast arc length builds up. When the derivative gets larger, the curve is steeper and the integrand increases. Differentiating the arc length function with the Fundamental Theorem of Calculus shows that its rate of change depends directly on the slope at that point. This connects local shape with total distance.

Arc length has practical uses when a path is curved. For example, if road barrier fencing is installed along a winding road, arc length helps measure the true ground distance. That prevents errors in materials, cost, and installation time.

Measuring Distance Along a Curve

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key terms and concepts

Learn how arc length uses small line segments, Riemann sums, and integrals to measure distance along a smooth curve and real paths like winding roads.

Measuring Distance Along a Curve | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Measuring Distance Along a Curve

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