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

定积分用于度量累积量，例如从速度-时间曲线计算总位移。某些性质可简化这些计算，包括可加性与常数倍性质。例如，一辆以不同速度行驶的汽车，其总位移由速度-时间曲线下的面积表示。当行程被划分为若干段时，定积分的可加性性质便具有了实用价值。假设汽车以连续的速度函数从 a 行驶到 c，然后从 c 行驶到 b。从 a 到 b 的总位移可通过将速度从 a 到 c 进行积分，然后加上从 c 到 b 的积分值来求得。这表明从 a 到 b 的定积分等于相邻区间上积分的和。另一个关键性质是常数倍性质。当汽车的速度被缩放时，此性质适用。当整个行程的初速度加倍时，总位移（曲线下的面积）也会加倍。这意味着常数与函数乘积的积分等于该常数乘以该函数的积分。

定积分的性质 ii

Properties of Definite Integral II

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定积分的性质 II

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Integrals

积分

KEY TERMS AND CONCEPTS

关键术语与概念

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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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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