Name: finding fish biomass with integrals
Uploaded: 2026-01-12T12:24:52.000Z
Duration: 1 min 27 s
Channel: JoVE

En un cuerpo de agua, el peso total de la población de peces se conoce como biomasa de peces. La tasa de crecimiento de una población de peces puede modelarse mediante la función G(t), donde t es el número de años y G(t) es la tasa de crecimiento medida en kilogramos por año.Se da la biomasa inicial para el año 2000.El objetivo es encontrar la biomasa de la población de peces en el año 2005.Para encontrar esto, la función de tasa se integra para encontrar la función de biomasa B(t).El método de sustitución se utiliza para resolver la integral tomando u y su derivada du.La integral resultante en u se simplifica, y la expresión se escribe de nuevo para obtener B(t) con una constante desconocida c.Las cantidades conocidas se toman como condiciones iniciales. Tras sustituir estos valores y resolver, se encuentra la constante desconocida c. Luego, sustituyendo esta c en la ecuación, se reescribe B(t).Para encontrar la biomasa en el año 2005, se sustituye t igual a 5 en esta ecuación de biomasa.Tras simplificar la ecuación, se encuentra la función de biomasa, que muestra la biomasa total de la población de peces en 2005.

modelos de crecimiento con integración: resolución de problemas

En el modelado de poblaciones, la integraci&#xF3;n proporciona un m&#xE9;todo sistem&#xE1;tico para determinar cantidades acumuladas a partir de tasas de cambio conocidas. Una de estas aplicaciones surge en ecolog&#xED;a, donde el peso total de una poblaci&#xF3;n de peces en un cuerpo de agua se denomina biomasa de la poblaci&#xF3;n de peces. Cuando la tasa de crecimiento de esta biomasa se conoce en funci&#xF3;n del tiempo, se puede usar el c&#xE1;lculo para determinar la biomasa total en una fecha posterior.
Tasa de Crecimiento y Funci&#xF3;n de Biomasa
Sea la tasa de crecimiento de la poblaci&#xF3;n de peces representada por la funci&#xF3;n G(t), donde t denota el tiempo medido en a&#xF1;os y G(t) representa la tasa de cambio de la biomasa en kilogramos por a&#xF1;o. La biomasa en s&#xED; se representa por la funci&#xF3;n B(t). Dado que la tasa de crecimiento describe c&#xF3;mo cambia la biomasa a lo largo del tiempo, la funci&#xF3;n de biomasa se obtiene integrando la tasa de crecimiento con respecto al tiempo.
Uso del M&#xE9;todo de Sustituci&#xF3;n
Para evaluar la integral que define la funci&#xF3;n de biomasa, se emplea el m&#xE9;todo de sustituci&#xF3;n. Esto implica seleccionar una variable de sustituci&#xF3;n adecuada u y su diferencial du para simplificar el integrando. Tras realizar la sustituci&#xF3;n, la integral se reescribe en t&#xE9;rminos de u, lo que facilita su evaluaci&#xF3;n. Una vez completada la integraci&#xF3;n, la expresi&#xF3;n se reexpresa en la variable original t, resultando en una forma general de la funci&#xF3;n de biomasa que incluye una constante de integraci&#xF3;n desconocida.
Aplicaci&#xF3;n de las Condiciones Iniciales
Se proporciona la biomasa inicial de la poblaci&#xF3;n de peces en el a&#xF1;o 2000, que sirve como condici&#xF3;n inicial. Sustituyendo el valor temporal y la biomasa correspondientes en la funci&#xF3;n general de biomasa, se puede determinar la constante de integraci&#xF3;n. Este paso garantiza que el modelo refleje con precisi&#xF3;n el estado conocido del sistema en el momento inicial. Tras encontrar esta constante, la funci&#xF3;n de biomasa se reescribe en su forma final.
Determinaci&#xF3;n de la Biomasa en 2005
Para calcular la biomasa de la poblaci&#xF3;n de peces en 2005, se establece la variable temporal cinco a&#xF1;os despu&#xE9;s del a&#xF1;o 2000. Sustituyendo este valor en la funci&#xF3;n de biomasa final y simplificando, se determina la biomasa total de la poblaci&#xF3;n de peces en 2005.

Modelos de crecimiento con integración: resolución de problemas

En el modelado de poblaciones, la integraci&#xF3;n proporciona un m&#xE9;todo sistem&#xE1;tico para determinar cantidades acumuladas a partir de tasas de ...

growth-models-with-integration-problem-solving

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Growth Models with Integration: Problem Solving

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 how biomass changes, the biomass function is found by integrating the growth rate with respect to time.

To evaluate the integral for B(t), the substitution method is used. A new variable, u, and its differential, du, are chosen to make the integrand simpler. After the integral is rewritten in terms of u, it is easier to solve. Then the expression is changed back into the original variable t, which gives a general biomass function with a constant of integration.

An initial biomass value from the year 2000 is then used to find that constant. This starting value acts as an initial condition. By substituting the known time and biomass into the general function, the constant of integration can be determined. The biomass function is then written in its final form.

To find the biomass in 2005, the time variable is set to five years after 2000. Substituting t = 5 into the final biomass function and simplifying gives the total fish biomass for that year.

Finding Fish Biomass with Integrals

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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

Use integration and substitution to model fish biomass from a growth rate, apply an initial condition, and calculate the biomass in 2005.

Finding Fish Biomass with Integrals | 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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Finding Fish Biomass with Integrals

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