Name: arc length formulas for line integrals
Uploaded: 2026-06-08T09:38:53.614Z
Duration: 1 min 25 s
Channel: JoVE

A mural is planned for the curved outer wall of a clock monument. To find the space available for the artwork, the total surface area of the curved wall must be found.The base of the wall follows a curved path, C. This curve is defined by parametric equations, where x and y depend on the parameter t. As t changes from a to b, it traces the curve from its starting point to its ending point.The height of the wall also changes from point to point along the curve and is given by a function of x and y.To find the total surface area, the wall is divided into n vertical strips along its curved base.Each strip is treated as a thin rectangle, with a width equal to a small arc length and a height equal to the wall&#x2019;s elevation. Multiplying these values gives the area of each small rectangle.Adding these small areas forms a Riemann sum. As the widths of these strips become infinitesimally small, the sum converges to a line integral.Evaluating this line integral using the curve length and the height function gives the total surface area of the wall. This gives the space available for the mural on the outer wall of the clock monument.

line integrals in the plane

Line integrals in the plane provide a method for evaluating quantities distributed along a curve, such as mass, work, or surface area. A curve C in the plane is commonly represented parametrically by x = x(t) and y = y(t), where the parameter t varies over an interval [a, b]. This representation allows geometric and physical quantities to be expressed in terms of a single variable, facilitating both analysis and computation.A line integral of a scalar function f(x, y) along a curve C is defined as the limit of a Riemann sum. The curve is partitioned into small segments, each with length &#x394;s. Over each segment, the function is approximately constant, so the contribution is f(xi,yi)&#x394;si. Summing these contributions and taking the limit yields\begin{equation*}\int_C f(x,y)\,ds\end{equation*}This integral represents the accumulation of the function f along the curve, weighted by arc length. Geometrically, it can describe quantities such as the mass of a wire with variable density or the area of a vertical surface with varying height.When the curve is parameterized, the arc length element is expressed as\begin{equation*}ds=\bm{\sqrt{\Biggl(\jfrac{\textit{d}\textit{x}}{\textit{d}\textit{t}}\Biggr)^2+\Biggl(\jfrac{\textit{d}\textit{y}}{\textit{d}\textit{t}}\Biggr)^2}}\,\textit{d}\textit{t}\end{equation*}Substituting into the line integral gives\begin{equation*}\int_C \textit{f}(\textit{x},\textit{y})\,ds=\int_a^b \textit{f}(\textit{x}(\textit{t}),\textit{y}(\textit{t}))\bm{\sqrt{\Biggl(\jfrac{\textit{dx}}{\textit{dt}}\Biggr)^2+\Biggl(\jfrac{\textit{dy}}{\textit{dt}}\Biggr)^2}}\,\textit{dt}\end{equation*}This formulation highlights how both the function and the curve&#x27;s geometry influence the result.Line integrals are widely used in applications involving distributed quantities. For example, in determining the surface area of a curved wall, the height function plays the role of f(x,y), while the curve defines the base. Similarly, in physics, line integrals describe work done by a force field along a path or the mass of a non-uniform wire.

Line Integrals in the Plane

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Vector Calculus and Theorems

向量微积分与定理

KEY TERMS AND CONCEPTS

关键术语与概念

平面上的线积分

arc length formulas for line integrals

Line integrals in the plane let us add up quantities along a curve, such as mass, work, or surface area. A curve C in the plane is usually written in parametric form as x = x(t) and y = y(t), with t moving over an interval [a, b]. This single-variable form makes it easier to study both the shape of the curve and the quantity spread along it.

For a scalar function f(x, y), a line integral along C is defined from a Riemann sum. The curve is split into small pieces with length Δs. On each piece, the function is treated as nearly constant, so the contribution is f(xi, yi)Δsi. Taking the limit gives the line integral ∫C f(x, y) ds.

This integral measures how the function accumulates along the curve, with arc length as the weight. It can be used to find the mass of a wire with variable density. It can also represent the area of a vertical surface with changing height.

When the curve is parameterized, the arc length element becomes ds = √[(dx/dt)^2 + (dy/dt)^2] dt. Substituting this into the line integral gives ∫C f(x, y) ds = ∫a^b f(x(t), y(t)) √[(dx/dt)^2 + (dy/dt)^2] dt. This shows that both the function and the geometry of the curve affect the final value.

Line integrals are useful in many applications that involve distributed quantities. In surface area problems, the height function plays the role of f(x, y), and the curve sets the base. In physics, line integrals describe work done by a force field along a path. They also describe the mass of a non-uniform wire.

Arc Length Formulas for Line Integrals

introduction to vector fields

Vector Fields in Everyday Motion

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gradient fields

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Line integrals in the plane let us add up quantities along a curve, such as mass, work, or surface area. A curve C in the plane is usually written in parametric form as x = x(t) and y = y(t), with t moving over an interval [a, b]. This single-variable form makes it easier to study both the shape of the curve and the quantity spread along it. For a scalar function f(x, y), a line integral along C is defined from a Riemann sum. The curve is split into small pieces with length Δs. On each piece,...

line integrals in space

Calculating Work Along Curved Paths

Line integrals in space are used to measure accumulation along a curved path in three dimensions. The path can be like a thin coiled spring, and it is described by a position vector r. That vector gives the x, y, and z coordinates in terms of one parameter t, which is often time or an angle. As t changes, r traces a smooth curve through space. For a scalar-valued function, the line integral with respect to arc length uses the magnitude of r'(t). Arc length is the distance measured along the...

applications of line integrals

Line Integrals in Work and Induction

Line integrals help measure work along a curved path and explain electromagnetic induction in a loop. In physics, they add up small contributions along a path when the force or electric field changes from point to point. For a particle moving on a curve, the work done by a force is written as W = ∫C F · dr. Here, F is the force and dr is a small displacement along the path. The dot product keeps only the part of the force that points along the motion. The same idea appears in electromagnetism...

the fundamental theorem for line integrals

the-fundamental-theorem-for-line-integrals

Path Independence in Line Integrals

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conservative vector fields

Testing Whether a Vector Field Is Conservative

A conservative vector field is a force field where the work done between two points depends only on the start and end points. Gravity is a simple example. For a ball in Earth’s gravitational field, the work depends on the change in height. That stays true even if the ball moves straight up or along a curved path. A vector field is conservative when it can be written as the gradient of a scalar potential function, f. In two dimensions, this is written as F(x,y) = = ∇f. The components come from...

green’s theorem

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extended versions of green’s theorem

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curl and divergence of vector fields

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vector forms of green’s theorem

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parametric surfaces

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tangent planes to a parametric surface

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surface integrals

Surface Area of Parametric Surfaces

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oriented surfaces

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surface integrals of vector fields: flux

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stokes’ theorem and its applications

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divergence theorem in 3d space

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vector calculus: problem solving

Curl of Gravitational Fields

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key terms and concepts

Learn how line integrals use arc length, parameterized curves, and Riemann sums to measure mass, work, and surface area in plane curves.

Arc Length Formulas for Line Integrals | 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

Arc Length Formulas for Line Integrals

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