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

Integrais definidas medem o acúmulo, como o deslocamento total de uma curva velocidade-tempo. Certas propriedades simplificam esses cálculos, incluindo a aditividade e as propriedades múltiplas constantes.Por exemplo, um carro viajando em velocidades variadas tem seu deslocamento total mostrado pela área sob a curva velocidade-tempo.Quando a viagem é dividida em segmentos, a propriedade de aditividade das integrais definidas torna-se útil.Suponha que o carro viaje de a a c, e depois de c a b, com uma função de velocidade contínua.O deslocamento total de a para b é determinado integrando a velocidade de a para c, e então somando a integral de c para b.Isso mostra que a integral definida de a a b é igual à soma das integrais nos intervalos adjacentes.Outra propriedade chave é a propriedade múltipla constante. Isso se aplica quando a velocidade do carro é escalada.Quando a velocidade original é dobrada durante toda a viagem, o deslocamento total — a área sob a curva — também é dobrado.Isso significa que a integral de uma constante vezes uma função é igual à constante multiplicada pela integral dessa função.

propriedades da integral definida ii

As integrais definidas s&#xE3;o ferramentas essenciais em c&#xE1;lculo, utilizadas para quantificar a varia&#xE7;&#xE3;o acumulada ao longo de um intervalo. Uma aplica&#xE7;&#xE3;o f&#xED;sica comum &#xE9; o c&#xE1;lculo do deslocamento total a partir de um gr&#xE1;fico de velocidade-tempo. Se uma fun&#xE7;&#xE3;o de velocidade, v(t), descreve o movimento de um objeto ao longo do tempo, a integral definida fornece o deslocamento l&#xED;quido entre os instantes a e b. Essa integral corresponde &#xE0; &#xE1;rea alg&#xE9;brica sob a curva de velocidade entre esses dois pontos.Duas propriedades fundamentais das integrais definidas auxiliam na simplifica&#xE7;&#xE3;o desses c&#xE1;lculos: a propriedade de aditividade e a propriedade do m&#xFA;ltiplo constante. A propriedade da aditividade afirma que, se uma fun&#xE7;&#xE3;o cont&#xED;nua for integrada em intervalos adjacentes, a soma dessas integrais &#xE9; igual &#xE0; integral sobre o intervalo total. Matematicamente, para 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>Isso &#xE9; particularmente &#xFA;til ao analisar o movimento em intervalos segmentados, como o percurso de um carro dividido em diferentes fases. Permite calcular o deslocamento por partes e somar os resultados.A propriedade do m&#xFA;ltiplo constante estabelece que multiplicar uma fun&#xE7;&#xE3;o por uma constante multiplica a integral pelo mesmo fator. Para uma 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>Essa propriedade &#xE9; especialmente relevante ao se considerar mudan&#xE7;as uniformes na velocidade &#x2014; como dobrar a velocidade ao longo da viagem &#x2014;, o que resulta em um aumento proporcional do deslocamento total. Juntas, essas propriedades simplificam a aplica&#xE7;&#xE3;o de integrais definidas em cen&#xE1;rios pr&#xE1;ticos, como a an&#xE1;lise cinem&#xE1;tica.

Propriedades da integral definida II

As integrais definidas s&#xE3;o ferramentas essenciais em c&#xE1;lculo, utilizadas para quantificar a varia&#xE7;&#xE3;o acumulada ao longo de um ...

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JoVE Core

Calculus

Cálculo

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Integrals

Integrais

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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...

distance problem

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...

definite integral

Riemann Sums and Area Estimates

Riemann sums and area estimates are a key part of calculus for finding the area under a curve. When a real-valued function is defined on a closed interval, the area under its graph can be approximated if an exact calculation is not easy to do. The interval is split into equal subintervals. Each subinterval becomes a rectangle. The rectangle’s width is the length of the subinterval, and its height comes from the function value at one chosen point inside that part of the interval. Multiplying...

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Gauss Sums for Riemann Rectangles

Riemann sums use rectangles to estimate the area under a curve in definite integration. The interval is split into subintervals, and the rectangle areas are added together. When those rectangle heights follow a regular pattern, sum formulas can make the calculation faster and more exact. This page focuses on sums that appear in arithmetic and polynomial sequences. The sum of consecutive integers, squares, and cubes is especially useful when the partition is uniform or when the data has a...

midpoint rule

Estimating Area with the Midpoint Rule

The Midpoint Rule estimates the area under a curve by using rectangles. It is a numerical method that helps when an exact area is hard to find or not practical to calculate. By using rectangle heights from midpoints, it gives a useful approximation over a chosen interval. To use the Midpoint Rule, the interval is split into equal subintervals. The midpoint of each subinterval is found by averaging its two endpoints. The function value at that midpoint gives the height of a rectangle over that...

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

The definite integral helps describe motion by showing how far an object travels over time. In this lesson, the key ideas are the Positivity Property and the Comparison Property of definite integrals. Both ideas connect integration to velocity and displacement in a way that is easy to interpret in real situations. The Positivity Property says that if an object’s velocity is nonnegative, then its total displacement over the time interval is also nonnegative. Nonnegative velocity means the...

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Definite Integrals: A Step-by-Step Strategy

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Accumulation and Rates in the Fundamental Theorem

The Fundamental Theorem of Calculus connects accumulation and rate of change. In this lesson, the idea is shown with a water tank that is filled by a pump. The pump rate changes over time, so the total water collected must be found by adding up the changing inflow. In water management, this kind of calculation matters for pressure control, operation timing, and system safety. The flow rate is given by a time-dependent function. To find how much water has entered the tank from the start to a...

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Definite Integrals with Antiderivatives

The Fundamental Theorem of Calculus, Part 2, connects definite integrals with antiderivatives. It shows that the area under a continuous curve can be found without adding infinitely many tiny rectangles one by one. Instead, the integral can be evaluated directly from an antiderivative. If a function f(x) is continuous on a closed interval [a, b], then its definite integral over that interval can be computed using any antiderivative F(x). The result is written as F(b) - F(a). This works because...

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Integration can be used to find the total volume of water in a tank when the inflow rate changes over time. In this example, the pump starts at 5 L/min. The inflow rate then increases by 2 L/min for each additional minute. That changing rate can be written as a linear function. To measure how much water has been added, the inflow rate function is integrated over time. The total water volume V(t) comes from this integral. When the power rule is used, the result includes both a quadratic term...

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Calculus for Reservoir Overflow Prediction

The Net Change Theorem links a function’s rate of change to its total accumulated change over an interval. In calculus, this means the definite integral of a derivative on [a,b] gives the net change in the original function. The theorem is useful in physics, economics, and engineering because it turns a rate into a total amount. One clear application is a reservoir that receives water from two inflow sources at the same time. The reservoir starts with 260 million cubic meters of water. Each...

substitution rule applied to indefinite integrals

substitution-rule-applied-to-indefinite-integrals

Work Done on a Nonlinear Spring

Work done on a nonlinear spring can be found by using integration and the substitution rule. A linear spring follows Hooke’s law, so its restoring force increases in direct proportion to displacement. That makes the force-displacement graph a straight line and the work calculation straightforward. A nonlinear spring does not obey Hooke’s law. Its restoring force changes with position in a non-proportional way, so the force-displacement graph is curved instead of linear. In the example shown,...

substitution rule applied to definite integrals

substitution-rule-applied-to-definite-integrals

Using Substitution to Evaluate Definite Integrals

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Finding Fish Biomass with Integrals

Fish biomass can be modeled with integration when the growth rate is known over time. In ecology, biomass means the total weight of a fish population in a body of water. If the growth rate of that biomass is given as a function of time, calculus can be used to find the total biomass at a later date. The growth rate is written as G(t), where t is time in years and G(t) is the rate of change of biomass in kilograms per year. The biomass itself is written as B(t). Because the growth rate shows...

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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