Name: definite integrals in motion analysis
Uploaded: 2026-01-12T12:24:47.000Z
Duration: 1 min 24 s
Channel: JoVE

Les intégrales définies mesurent l’accumulation, comme le déplacement total d’une courbe vitesse-temps. Certaines propriétés simplifient ces calculs, notamment l’additivité et les propriétés multiples constantes.Par exemple, une voiture se déplaçant à différentes vitesses a son déplacement total indiqué par la zone sous la courbe vitesse-temps.Lorsque le trip est divisé en segments, la propriété d’additivité des intégrales définies devient utile.Supposons que la voiture se déplace de a à c, puis de c à b, avec une fonction de vitesse continue.Le déplacement total de a à b est déterminé en intégrant la vitesse de a à c, puis en additionnant l’intégrale de c à b.Cela montre que l’intégrale définie de a à b est égale à la somme des intégrales sur les intervalles adjacents.Une autre propriété clé est la propriété du multiple constant. Cela s’applique lorsque la vitesse de la voiture est mise à l’échelle.Lorsque la vitesse initiale est doublée sur l’ensemble du trajet, le déplacement total — la surface sous la courbe — est également doublé.Cela signifie que l’intégrale d’une constante multipliée par une fonction est égale à la constante multipliée par l’intégrale de cette fonction.

propriétés de l’intégrale définie ii

Les int&#xE9;grales d&#xE9;finies sont des outils essentiels en analyse, utilis&#xE9;s pour quantifier une variation cumul&#xE9;e sur un intervalle. Une application physique courante consiste &#xE0; calculer le d&#xE9;placement total &#xE0; partir d&#x27;un graphique vitesse-temps. Si une fonction vitesse, v(t), d&#xE9;crit le mouvement d&#x27;un objet au cours du temps, l&#x27;int&#xE9;grale d&#xE9;finie donne le d&#xE9;placement net entre les instants a et b. Cette int&#xE9;grale correspond &#xE0; l&#x2019;aire alg&#xE9;brique sous la courbe de la vitesse entre ces deux points.Deux propri&#xE9;t&#xE9;s fondamentales des int&#xE9;grales d&#xE9;finies permettent de simplifier de tels calculs : la propri&#xE9;t&#xE9; d&#x2019;additivit&#xE9; et la propri&#xE9;t&#xE9; du facteur constant. La propri&#xE9;t&#xE9; d&#x2019;additivit&#xE9; stipule que si une fonction continue est int&#xE9;gr&#xE9;e sur des intervalles adjacents, la somme de ces int&#xE9;grales est &#xE9;gale &#xE0; l&#x2019;int&#xE9;grale sur l&#x2019;intervalle total. Math&#xE9;matiquement, pour a &lt; c &lt; b,<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:35.56%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/42035/42035_Equation_1_editor_42035.svg" alt="42035_Equation_1.svg" /></figure>C&#x2019;est particuli&#xE8;rement utile pour analyser un mouvement sur des intervalles segment&#xE9;s, comme le trajet d&#x27;une voiture divis&#xE9; en diff&#xE9;rentes phases. Cela permet de calculer le d&#xE9;placement par morceaux et d&#x2019;additionner les r&#xE9;sultats.La propri&#xE9;t&#xE9; de facteur constant exprime que si l&#x2019;on multiplie une fonction par une constante, alors son int&#xE9;grale est multipli&#xE9;e par le m&#xEA;me facteur. Pour une constante, c,<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:35.56%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/42035/42035_Equation_2_editor_42035.svg" alt="42035_Equation_2.svg" /></figure>Cette propri&#xE9;t&#xE9; est particuli&#xE8;rement pertinente dans le cas o&#xF9; l&#x27;on consid&#xE8;re des changements uniformes de vitesse &#x2014; comme le fait de doubler la vitesse tout au long du trajet &#x2014; ce qui entra&#xEE;ne une augmentation proportionnelle du d&#xE9;placement total. Ensemble, ces propri&#xE9;t&#xE9;s simplifient l&#x27;application des int&#xE9;grales d&#xE9;finies dans des situations pratiques, comme l&#x2019;analyse cin&#xE9;matique.

Propriétés de l’intégrale définie II

Les int&#xE9;grales d&#xE9;finies sont des outils essentiels en analyse, utilis&#xE9;s pour quantifier une variation cumul&#xE9;e sur un intervalle. Une ...

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JoVE Core

Calculus

Calcul

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Integrals

Intégrales

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

Properties of Definite Integral II

definite integrals in motion analysis

Definite integrals can be used to find displacement from a velocity-time graph. A velocity function v(t) describes how motion changes over time. The definite integral of v(t) from a to b gives the net displacement between those times. It also represents the signed area under the velocity curve between the two points.

Two useful properties make definite integrals easier to work with in motion problems. The additivity property says that if a continuous function is integrated over adjacent intervals, the separate integrals can be added to get the integral over the full interval. For a &lt; c &lt; b, the total from a to b equals the sum from a to c and from c to b. This is helpful when a trip has more than one phase, such as a car moving through different sections of a journey.

The constant multiple property gives another shortcut. If a function is multiplied by a constant c, then its definite integral is also multiplied by the same constant. In motion, this matters when the velocity changes by the same factor at every point in time, such as doubling the speed throughout a trip. The total displacement then increases by the same proportion. Together, these properties make definite integrals easier to apply in kinematics.

Definite Integrals in Motion Analysis

area problem

Approximating Area Under a Curve

Approximating area under a curve is a key way to measure regions with curved boundaries. Straight-sided shapes are easy to handle with formulas for rectangles, triangles, and polygons. Curved regions need a different method because standard geometry does not work well on them. The area problem solves this by breaking the region into smaller, simpler shapes. A common approach uses rectangles to estimate the space under a function. Each rectangle gets its height from the function at a chosen...

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Estimating Distance from Changing Velocity

Estimating distance from changing velocity uses small time intervals and velocity values. When an object speeds up or slows down, its total displacement can be found by adding many short pieces of motion. This method gives an approximation first, and integral calculus gives the exact result later. A runner who accelerates during the first three seconds of a race shows how the idea works. Velocity can be measured every half-second and multiplied by the 0.5-second time step. Using left-endpoint...

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Riemann Sums and Area Estimates

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Gauss Sums for Riemann Rectangles

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Estimating Area with the Midpoint Rule

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properties of definite integral i

Definite Integrals for Car Motion

Definite integrals help describe a car’s motion over time by linking velocity to displacement. In a velocity-time graph, the definite integral shows how velocity builds up over a chosen time interval. The area under the curve gives the total displacement, so the graph becomes a clear picture of how far the car moves. One key idea is linearity of definite integrals. This property lets you add or subtract velocity functions while keeping their accumulated effect. If a car’s motion is made of...

properties of definite integral ii

Definite integrals can be used to find displacement from a velocity-time graph. A velocity function v(t) describes how motion changes over time. The definite integral of v(t) from a to b gives the net displacement between those times. It also represents the signed area under the velocity curve between the two points. Two useful properties make definite integrals easier to work with in motion problems. The additivity property says that if a continuous function is integrated over adjacent...

properties of definite integral iii

Definite Integral Rules for Motion

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Definite Integrals: A Step-by-Step Strategy

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Work Done on a Nonlinear Spring

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Using Substitution to Evaluate Definite Integrals

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key terms and concepts

Learn how definite integrals find net displacement from a velocity-time graph and how additivity and constant multiple properties simplify motion problems.

Definite Integrals in Motion Analysis | 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

Definite Integrals in Motion Analysis

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