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

Consideremos una función sobre un intervalo cerrado. Para aproximar el área bajo la curva, dividir el intervalo en subintervalos iguales, formando rectángulos de igual anchura.Selecciona un punto arbitrario dentro del subintervalo y utiliza el valor de la función ahí como la altura del rectángulo. Multiplicar la altura por el ancho da el área de ese rectángulo. El área total bajo la curva se aproxima sumando las áreas de todos los rectángulos y se denomina suma de Riemann.La integral definida de una función sobre un intervalo dado se define como el límite de su suma de Riemann a medida que el número de subintervalos tiende al infinito y el ancho de cada rectángulo se vuelve infinitesimalmente pequeño, siempre que este límite exista.Por ejemplo, imaginemos un coche que viaja a una velocidad variable. Divide el tiempo total de viaje en intervalos iguales.En cada intervalo, selecciona un punto temporal arbitrario y toma la velocidad correspondiente. Multiplica la velocidad por la anchura del intervalo para estimar la distancia recorrida.La suma de estos forma una suma de Riemann, y en este ejemplo, el área bajo la curva se interpreta físicamente como la distancia recorrida por el coche. Cuando la velocidad sigue siendo positiva, la distancia total se determina integrando la función de velocidad dentro de los límites.

integral definida

Definite Integral

Considere una funci&#xF3;n real definida en un intervalo cerrado. Uno de los objetivos fundamentales del c&#xE1;lculo es determinar el &#xE1;rea bajo la gr&#xE1;fica de dicha funci&#xF3;n. Cuando no se dispone de un c&#xE1;lculo exacto, esta &#xE1;rea puede estimarse dividiendo el intervalo en un n&#xFA;mero finito de subintervalos iguales. Cada subintervalo corresponde a un rect&#xE1;ngulo cuyo ancho es la longitud del subintervalo y cuya altura est&#xE1; determinada por el valor de la funci&#xF3;n en un punto seleccionado dentro de ese subintervalo.
Sumas de Riemann y aproximaci&#xF3;n del &#xE1;rea
Para cada subintervalo, se elige un punto arbitrario y el valor de la funci&#xF3;n en ese punto se toma como la altura del rect&#xE1;ngulo. El &#xE1;rea de cada rect&#xE1;ngulo se obtiene multiplicando su altura por su ancho. Sumar las &#xE1;reas de todos los rect&#xE1;ngulos proporciona una aproximaci&#xF3;n al &#xE1;rea total bajo la curva. Esta aproximaci&#xF3;n se conoce como suma de Riemann. A medida que aumenta el n&#xFA;mero de subintervalos y disminuye su anchura, la aproximaci&#xF3;n, por lo general, se vuelve m&#xE1;s precisa, reflejando con mayor precisi&#xF3;n la forma de la curva.
Integral definida como proceso l&#xED;mite
La integral definida refina el concepto de suma de Riemann al considerar qu&#xE9; sucede cuando la partici&#xF3;n del intervalo se vuelve infinitamente fina. La integral definida de una funci&#xF3;n en un intervalo dado se define como el l&#xED;mite al que tienden las sumas de Riemann a medida que el n&#xFA;mero de rect&#xE1;ngulos crece sin l&#xED;mite y la anchura de cada rect&#xE1;ngulo tiende a cero, siempre que exista este valor l&#xED;mite. En este sentido, la integral definida representa el &#xE1;rea exacta bajo la curva en el intervalo especificado.
Interpretaci&#xF3;n en contextos f&#xED;sicos
Una ilustraci&#xF3;n pr&#xE1;ctica de este concepto surge en el estudio del movimiento. Considere un autom&#xF3;vil que recorre una trayectoria recta con una velocidad que var&#xED;a con el tiempo. Dividiendo el tiempo total de viaje en segmentos iguales y utilizando la velocidad en un instante elegido dentro de cada segmento, se puede estimar la distancia recorrida durante ese segmento. La suma de estas distancias produce una suma de Riemann que se aproxima a la distancia total recorrida. Cuando la velocidad se mantiene positiva durante todo el intervalo, la integral definida de la funci&#xF3;n de velocidad corresponde a la distancia total recorrida por el autom&#xF3;vil.

Integral definida

Considere una funci&#xF3;n real definida en un intervalo cerrado. Uno de los objetivos fundamentales del c&#xE1;lculo es determinar el &#xE1;rea bajo la ...

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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.

calculus

Riemann Sums and Area Estimates

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