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

Definite integrals measure accumulation, such as total displacement from a velocity-time curve. Certain properties simplify these calculations, including additivity and the constant multiple properties.For example, a car traveling at varying speeds has its total displacement shown by the area under the velocity-time curve.When the trip is divided into segments, the additivity property of definite integrals becomes useful.Suppose the car travels from a to c, and then from c to b, with a continuous velocity function.The total displacement from a to b is found by integrating the velocity from a to c, and then adding the integral from c to b.This shows that the definite integral from a to b equals the sum of the integrals over the adjacent intervals.Another key property is the constant multiple property. This applies when the car&#8217;s velocity is scaled.When the original velocity is doubled for the whole trip, the total displacement&#8212;the area under the curve&#8212;is also doubled.This means the integral of a constant times a function equals the constant multiplied by the integral of that function.

properties of definite integral ii

Definite integrals are essential tools in calculus, used to quantify accumulated change over an interval. A common physical application is calculating the total displacement from a velocity-time graph. If a velocity function, v(t), describes the motion of an object over time, the definite integral gives the net displacement between times a and b. This integral corresponds to the signed area under the velocity curve between those two points.Two fundamental properties of definite integrals aid in simplifying such calculations: the additivity property and the constant multiple property. The additivity property states that if a continuous function is integrated over adjacent intervals, the sum of these integrals equals the integral over the combined interval. Mathematically, for a &#60; c &#60; b,<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:33.22%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/42035/42035_Equation_1_editor_42035.svg" alt="Integral equation for interval splitting, showing integration from a to b; mathematical formula." data-altupdatedat="1775223722000" data-alt="42035_Equation_1.svg" /></figure>This is particularly useful when analyzing motion over segmented intervals, such as a car&#8217;s journey divided into different phases. It allows one to compute displacement piecewise and sum the results.The constant multiple property expresses that scaling a function by a constant scales the integral by the same factor. For a constant, c,<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:33.22%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/42035/42035_Equation_2_editor_42035.svg" alt="Integral properties, equation, demonstrating constant factor rule, mathematical theorem." data-altupdatedat="1775223722000" data-alt="42035_Equation_2.svg" /></figure>This property is especially relevant when considering uniform changes in velocity&#8212;such as doubling the speed throughout the trip&#8212;which results in a proportional increase in total displacement. Together, these properties streamline the application of definite integrals in practical scenarios, such as kinematic analysis.

Properties of Definite Integral II

Definite integrals are essential tools in calculus, used to quantify accumulated change over an interval. A common physical application is calculating the ...

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Calculus

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Integrals

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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

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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

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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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