Name: riemann sums and area estimates
Uploaded: 2026-01-12T12:24:23.000Z
Duration: 1 min 29 s
Channel: JoVE

Considere uma função sobre um intervalo fechado. Para aproximar a área sob a curva, divida-se o intervalo em subintervalos iguais, formando retângulos de largura igual.Selecione um ponto arbitrário dentro do subintervalo e use o valor da função ali como a altura do retângulo. Multiplicando a altura pela largura, obtém-se a área desse retângulo. A área total sob a curva é aproximada somando as áreas de todos os retângulos e é chamada de soma de Riemann.A integral definida de uma função sobre um dado intervalo é definida como o limite de sua soma de Riemann à medida que o número de subintervalos tende ao infinito e a largura de cada retângulo se torna infinitesimalmente pequena, desde que esse limite exista.Por exemplo, considere um carro viajando com velocidades variadas. Divida o tempo total de viagem em intervalos de tempo iguais.Em cada intervalo, selecione um ponto de tempo arbitrário e tome a velocidade correspondente. Multiplique a velocidade pela largura do intervalo para estimar a distância percorrida.A soma desses forma uma soma de Riemann e, neste exemplo, a área sob a curva é fisicamente interpretada como a distância percorrida pelo carro. Quando a velocidade permanece positiva, a distância total é encontrada integrando a função de velocidade dentro dos limites.

integral definida

Considere uma fun&#xE7;&#xE3;o de valores reais definida em um intervalo fechado. Um dos objetivos fundamentais do c&#xE1;lculo &#xE9; determinar a &#xE1;rea sob o gr&#xE1;fico dessa fun&#xE7;&#xE3;o. Quando um c&#xE1;lculo exato n&#xE3;o est&#xE1; prontamente dispon&#xED;vel, essa &#xE1;rea pode ser estimada dividindo-se o intervalo em um n&#xFA;mero finito de subintervalos iguais. Cada subintervalo corresponde a um ret&#xE2;ngulo cuja largura &#xE9; o comprimento do subintervalo e cuja altura &#xE9; determinada pelo valor da fun&#xE7;&#xE3;o em um ponto selecionado dentro desse subintervalo.
Somas de Riemann e Aproxima&#xE7;&#xE3;o da &#xC1;rea
Para cada subintervalo, um ponto arbitr&#xE1;rio &#xE9; escolhido, e o valor da fun&#xE7;&#xE3;o nesse ponto &#xE9; usado como a altura do ret&#xE2;ngulo. A &#xE1;rea de cada ret&#xE2;ngulo &#xE9; obtida multiplicando-se sua altura por sua largura. A soma das &#xE1;reas de todos os ret&#xE2;ngulos fornece uma aproxima&#xE7;&#xE3;o da &#xE1;rea total sob a curva. Essa aproxima&#xE7;&#xE3;o &#xE9; conhecida como soma de Riemann. &#xC0; medida que o n&#xFA;mero de subintervalos aumenta e suas larguras diminuem, a aproxima&#xE7;&#xE3;o geralmente se torna mais precisa, refletindo mais fielmente a forma da curva.
Integral Definida como um Processo Limite
A integral definida refina a ideia de uma soma de Riemann ao considerar o que acontece quando a parti&#xE7;&#xE3;o do intervalo se torna infinitamente fina. A integral definida de uma fun&#xE7;&#xE3;o em um intervalo dado &#xE9; definida como o valor ao qual as somas de Riemann se aproximam quando o n&#xFA;mero de ret&#xE2;ngulos cresce indefinidamente e a largura de cada ret&#xE2;ngulo se torna desprez&#xED;vel, desde que esse valor limite exista. Nesse sentido, a integral definida representa a &#xE1;rea exata sob a curva no intervalo especificado.
Interpreta&#xE7;&#xE3;o em Contextos F&#xED;sicos
Uma ilustra&#xE7;&#xE3;o pr&#xE1;tica desse conceito surge no estudo do movimento. Considere um carro viajando em linha reta com uma velocidade que varia ao longo do tempo. Dividindo o tempo total de viagem em segmentos iguais e usando a velocidade em um instante escolhido dentro de cada segmento, a dist&#xE2;ncia percorrida durante esse segmento pode ser estimada. A soma dessas dist&#xE2;ncias resulta em uma soma de Riemann que aproxima a dist&#xE2;ncia total percorrida. Quando a velocidade permanece positiva ao longo do intervalo, a integral definida da fun&#xE7;&#xE3;o de velocidade corresponde &#xE0; dist&#xE2;ncia total percorrida pelo carro.

Integral Definida

Considere uma fun&#xE7;&#xE3;o de valores reais definida em um intervalo fechado. Um dos objetivos fundamentais do c&#xE1;lculo &#xE9; determinar a ...

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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 height by width gives the area of each rectangle.

Adding the areas of all the rectangles gives a Riemann sum. This sum is an approximation of the total area under the curve. As the number of subintervals increases and the widths get smaller, the estimate usually becomes more accurate because it follows the shape of the curve more closely.

The definite integral builds on this idea by taking the limit of the Riemann sums. It looks at what happens as the partition of the interval becomes infinitely fine and the rectangle widths become negligibly small, provided that the limit exists. In that case, the definite integral gives the exact area under the curve over the interval.

This idea also has a physical meaning in motion. A car moving along a straight path with changing velocity can be studied by dividing the travel time into equal parts. Using the velocity at a chosen time in each part gives an estimate for the distance traveled in that segment. Adding those distances gives a Riemann sum for total distance, and when the velocity stays positive, the definite integral of velocity over time matches the total distance covered.

Riemann Sums and Area Estimates

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key terms and concepts

Learn how Riemann sums use rectangles to estimate area under a curve and how the definite integral gives the exact value as the intervals shrink.

Riemann Sums and Area Estimates | 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.

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Riemann Sums and Area Estimates

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