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

Funkcja długości łuku pokazuje całkowitą odległość pokonaną wzdłuż gładkiej krzywej od stałego punktu startowego do zmiennego końca.Dla ciągłej i różniczkowalnej krzywej otrzymuje się to, sumując małe odcinki liniowe wzdłuż krzywej. Te segmenty przybliżają krzywą za pomocą zmian poziomych i pionowych, podobnie jak suma Riemanna.Gdy rozmiar segmentu zbliża się do zera, suma staje się całką dająca dokładną długość łuku.Aby wyrazić długość łuku jako funkcję, wewnątrz całki używa się zmiennej fikcyjnej, co pozwala na zmienność górnej granicy.Całka zawiera pierwiastek kwadratowy z jeden plus kwadrat pochodnej. Zawsze jest większa lub równa jeden i rośnie wraz ze stromością krzywej, co powoduje szybszy wzrost długości łuku.Wykorzystanie Fundamentalnego Twierdzenia Rachunku różniczkowego do różniczkowania funkcji daje szybkość zmiany długości łuku, która zależy bezpośrednio od nachylenia krzywej.Na przykład podczas montażu ogrodzenia drogowego wzdłuż krętej drogi funkcja długości łuku dokładnie mierzy odległość gruntu, co pomaga uniknąć niedoszacowania materiałów, kosztów i czasu montażu.

funkcja długości łuku

Arc Length Function

Funkcja d&#x142;ugo&#x15B;ci &#x142;uku przedstawia ca&#x142;kowit&#x105; odleg&#x142;o&#x15B;&#x107; przebyt&#x105; wzd&#x142;u&#x17C; g&#x142;adkiej krzywej, mierzon&#x105; od ustalonego punktu pocz&#x105;tkowego do zmiennego punktu ko&#x144;cowego. W przypadku krzywych ci&#x105;g&#x142;ych i r&#xF3;&#x17C;niczkowalnych d&#x142;ugo&#x15B;&#x107; &#x142;uku zapewnia precyzyjny spos&#xF3;b kwantyfikacji odleg&#x142;o&#x15B;ci, gdy przybli&#x17C;enia liniowe s&#x105; niewystarczaj&#x105;ce.Aby wyznaczy&#x107; d&#x142;ugo&#x15B;&#x107; &#x142;uku, krzywa jest dzielona na wiele ma&#x142;ych odcink&#xF3;w. Ka&#x17C;dy odcinek jest przybli&#x17C;any lini&#x105; prost&#x105;, kt&#xF3;rej d&#x142;ugo&#x15B;&#x107; zale&#x17C;y od zmian poziomych i pionowych w tym przedziale. Te liniowe odcinki przypominaj&#x105; struktur&#x119; sumy Riemanna. Wraz ze wzrostem liczby odcink&#xF3;w i zmniejszaniem si&#x119; ich szeroko&#x15B;ci w kierunku zera przybli&#x17C;enie zbiega do ca&#x142;ki, kt&#xF3;ra daje dok&#x142;adn&#x105; d&#x142;ugo&#x15B;&#x107; krzywej.Dla funkcji y = f(x) r&#xF3;&#x17C;niczkowalnej na przedziale d&#x142;ugo&#x15B;&#x107; &#x142;uku od ustalonego punktu x = a do zmiennego punktu ko&#x144;cowego x jest dana wzorem:\begin{equation*}L(x) = \int_a^x \bm{\sqrt{1 + (f&#x27;(u))^2}}\, du\end{equation*}Funkcja podca&#x142;kowa jest zawsze wi&#x119;ksza lub r&#xF3;wna jednemu, co odzwierciedla fakt, &#x17C;e najkr&#xF3;tsz&#x105; odleg&#x142;o&#x15B;ci&#x105; mi&#x119;dzy dwoma punktami jest linia prosta. Wraz ze wzrostem modu&#x142;u pochodnej, wskazuj&#x105;c na bardziej strom&#x105; krzyw&#x105;, ro&#x15B;nie warto&#x15B;&#x107; funkcji podca&#x142;kowej, co powoduje szybsze narastanie d&#x142;ugo&#x15B;ci &#x142;uku.R&#xF3;&#x17C;nicowanie funkcji d&#x142;ugo&#x15B;ci &#x142;uku za pomoc&#x105; twierdzenia podstawowego rachunku ca&#x142;kowego pokazuje, &#x17C;e jej tempo zmian w dowolnym punkcie zale&#x17C;y bezpo&#x15B;rednio od nachylenia krzywej w tym punkcie. Podkre&#x15B;la to &#x15B;cis&#x142;y zwi&#x105;zek mi&#x119;dzy lokalnym zachowaniem geometrycznym a ca&#x142;kowit&#x105; odleg&#x142;o&#x15B;ci&#x105; nagromadzon&#x105;.Funkcje d&#x142;ugo&#x15B;ci &#x142;uku maj&#x105; kluczowe znaczenie w zastosowaniach praktycznych, gdzie konieczny jest precyzyjny pomiar odleg&#x142;o&#x15B;ci wzd&#x142;u&#x17C; zakrzywionych &#x15B;cie&#x17C;ek i tor&#xF3;w. Na przyk&#x142;ad podczas monta&#x17C;u ogrodzenia drogowego wzd&#x142;u&#x17C; kr&#x119;tej drogi obliczenia d&#x142;ugo&#x15B;ci &#x142;uku gwarantuj&#x105; pomiar rzeczywistej odleg&#x142;o&#x15B;ci w terenie, zapobiegaj&#x105;c niedoszacowaniu materia&#x142;&#xF3;w, koszt&#xF3;w i czasu monta&#x17C;u.

Funkcja długości łuku

Funkcja d&#x142;ugo&#x15B;ci &#x142;uku przedstawia ca&#x142;kowit&#x105; odleg&#x142;o&#x15B;&#x107; przebyt&#x105; wzd&#x142;u&#x17C; g&#x142;adkiej ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Application of Techniques of Integration

Zastosowanie technik całkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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

arc length of a curve

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arc length function

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...

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applications of integration to find hydrostatic pressure

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Hydrostatic Force on Dam Walls with Integration

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applications of integration to find centers of mass

applications-of-integration-to-find-centers-of-mass

Center of Mass in Discrete and Continuous Systems

Center of mass can be found from rotational equilibrium and from integration. A balanced plank on a pivot shows the idea clearly. When the net torque about the pivot is zero, the system is in balance. Torque is the force on an object times its perpendicular distance from the pivot. At that point, the pivot lies at the center of mass. For a set of discrete point masses, the center of mass is the position where the masses balance by weight. Each mass contributes its mass times its position,...

theorem of pappus

Pappus Theorem for Volume and Surface Area

Pappus Theorem connects centroid motion to the volume and surface area of solids of revolution. It is also called the Pappus–Guldinus Theorem. The method uses geometry to find these values when a plane region or a plane curve is rotated about an external axis. The theorem has two related parts. The first part gives the volume of a solid formed by rotating a plane region around an axis outside the region. In this case, the volume equals the area of the region multiplied by the distance traveled...

applications of integration to find consumer surplus

applications-of-integration-to-find-consumer-surplus

Consumer Surplus from Demand and Integration

Consumer surplus measures the economic gain buyers receive when they pay less than the highest price they are willing to pay. In microeconomics, this idea comes from the demand function, which links quantity to the price consumers will pay. Because willingness to pay usually falls as quantity rises, the demand curve slopes downward. When a product is sold at a fixed price, some buyers would have paid more for the first units they bought. The difference between that willingness to pay and the...

applications of integration to find blood flow

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Using Integration to Model Blood Flow

Blood flow in a cylindrical blood vessel can be modeled with integration and laminar flow. Laminar flow is smooth motion in parallel layers. In this model, blood speed changes with distance from the center of the vessel. The blood moves fastest along the central axis. As the distance from the center increases, the velocity decreases. At the vessel wall, the velocity drops to zero because of viscous drag. This changing speed across the cross-section is the key idea behind the calculation. To...

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Normal distribution helps explain how standardized test scores can be spread across a group. It is a key idea in statistics for describing symmetry, center, and variability in large sets of data. Test score data often start as a histogram. In a histogram, each bar shows how many scores fall within a score range. To turn that count-based graph into a continuous probability model, the histogram is normalized by dividing each bar height by the total number of observations and by the bin width. <br...

applications of integration to probability density functions

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Finding Probability with Area Under a PDF

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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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