Name: riemann sum area estimates
Uploaded: 2026-01-12T12:24:45.000Z
Duration: 1 min 24 s
Channel: JoVE

La integración aproximada se utiliza cuando no se puede calcular el valor exacto de una integral definida.Esto suele ocurrir en dos casos principales: cuando la antiderivada de la función es desconocida o no existe en forma cerrada, y cuando la función está compuesta por datos empíricos, como un conjunto de puntos discretos de un experimento, en lugar de una fórmula continua.En tales situaciones, las integrales definidas se estiman usando sumas de Riemann, que dividen el intervalo en n subintervalos iguales de anchura Δx.Para cada subintervalo se construye un rectángulo, cuya altura está determinada por el valor de la función en un punto específico dentro de ese subintervalo.En la aproximación del extremo izquierdo Ln, la altura de cada rectángulo se establece por el valor de la función en el extremo izquierdo.Si la función está aumentando, este método subestima el área; si la función disminuye, sobreestima.La aproximación del extremo derecho Rn utiliza el extremo derecho de cada subintervalo. Este método sobreestima el área si la función está aumentando y viceversa.Estos métodos utilizan geometría básica —añadiendo áreas rectangulares— para estimar integrales de funciones complejas o desconocidas.

integración aproximada

En muchos contextos pr&#xE1;cticos y te&#xF3;ricos, el valor exacto de una integral definida puede ser inaccesible. Esta limitaci&#xF3;n suele surgir cuando la primitiva de una funci&#xF3;n es desconocida o no puede expresarse en una forma matem&#xE1;tica cerrada. Alternativamente, tambi&#xE9;n sucede cuando una funci&#xF3;n no se define mediante una f&#xF3;rmula, sino mediante un conjunto finito de datos emp&#xED;ricos, como los recopilados durante experimentos. En estos casos, las t&#xE9;cnicas de integraci&#xF3;n aproximada ofrecen una soluci&#xF3;n valiosa.
Uno de los enfoques m&#xE1;s utilizados es el uso de sumas de Riemann, que estiman el &#xE1;rea bajo una curva subdividiendo el intervalo de integraci&#xF3;n en varias partes iguales. Cada subintervalo se asocia con un rect&#xE1;ngulo cuya altura est&#xE1; determinada por el valor de la funci&#xF3;n en un punto elegido dentro de ese subintervalo. Esta sencilla interpretaci&#xF3;n geom&#xE9;trica permite obtener aproximaciones num&#xE9;ricas cuando los m&#xE9;todos anal&#xED;ticos no son viables.
La elecci&#xF3;n del punto dentro de cada subintervalo da lugar a diferentes m&#xE9;todos. En la aproximaci&#xF3;n del extremo izquierdo, la altura de cada rect&#xE1;ngulo se basa en el valor de la funci&#xF3;n en el extremo izquierdo del subintervalo correspondiente. Cuando la funci&#xF3;n es creciente, este m&#xE9;todo tiende a subestimar el &#xE1;rea total; cuando es decreciente, suele sobreestimarla. Por el contrario, la aproximaci&#xF3;n del extremo derecho utiliza el valor de la funci&#xF3;n en el extremo derecho de cada subintervalo, lo que da lugar a una sobreestimaci&#xF3;n para las funciones que son crecientes y una subestimaci&#xF3;n para las funciones que son decrecientes.
Estas t&#xE9;cnicas, aunque elementales en su formulaci&#xF3;n, son herramientas fundamentales en el an&#xE1;lisis num&#xE9;rico. Son especialmente valiosas en aplicaciones cient&#xED;ficas y de ingenier&#xED;a donde las integrales deben evaluarse a partir de mediciones discretas o para funciones demasiado complejas para la integraci&#xF3;n simb&#xF3;lica.

Integración Aproximada

En muchos contextos pr&#xE1;cticos y te&#xF3;ricos, el valor exacto de una integral definida puede ser inaccesible. Esta limitaci&#xF3;n suele surgir ...

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Approximate Integration

riemann sum area estimates

Riemann sums estimate the area under a curve when an exact definite integral is hard to find. This often happens when a function has no known antiderivative or cannot be written in a closed form. It also happens when the function comes from measured data instead of a formula.

The method works by splitting the interval of integration into equal parts. Each subinterval gets a rectangle, and the rectangle height comes from the function value at one chosen point in that part. Adding the rectangle areas gives an approximate integration result.

A left endpoint approximation uses the function value at the left end of each subinterval. For an increasing function, this method usually underestimates the total area. For a decreasing function, it usually overestimates it.

A right endpoint approximation uses the function value at the right end of each subinterval. For an increasing function, it usually overestimates the area. For a decreasing function, it usually underestimates it.

These rectangle-based estimates are simple, but they are useful in numerical analysis. They are especially helpful in science and engineering when an integral must be found from discrete measurements or when symbolic integration is not practical.

Riemann Sum Area Estimates

integration by parts: indefinite integrals

integration-by-parts-indefinite-integrals

Integration by Parts in AC Circuit Analysis

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integration by parts: definite integrals

Using Parts and Substitution for Area

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rationalizing substitutions

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approximate integration

Riemann sums estimate the area under a curve when an exact definite integral is hard to find. This often happens when a function has no known antiderivative or cannot be written in a closed form. It also happens when the function comes from measured data instead of a formula. The method works by splitting the interval of integration into equal parts. Each subinterval gets a rectangle, and the rectangle height comes from the function value at one chosen point in that part. Adding the rectangle...

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Estimating Area with Simpson’s Rule

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improper integrals: discontinuous integrands

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Improper Integrals at an Endpoint

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key terms and concepts

Learn how Riemann sums estimate area under a curve using left and right endpoint rectangles when exact integration is not available.

Riemann Sum 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.

calculus

Riemann Sum Area Estimates

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