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

In a water body, the total weight of the fish population is known as the biomass of fish. The rate of growth of a fish population can be modeled by the function G(t), where t is the number of years and G(t) is the growth rate measured in kilograms per year.The initial biomass for the year 2000 is given.The goal is to find the biomass of the fish population in the year 2005.To find this, the rate function is integrated to find the biomass function B(t).The substitution method is used to solve the integral by taking u and its derivative du.The resulting integral in u is simplified, and the expression is then written back to get B(t) with an unknown constant c.The known quantities are taken as the initial conditions. After substituting these values and solving, the unknown constant c is found. Then, substituting this c into the equation, B(t) is rewritten.To find the biomass in the year 2005, t equals 5 is substituted into this biomass equation.After simplifying the equation, the biomass function is found, showing the total biomass of the fish population in 2005.

groeimodellen met integratie: probleemoplossing

In de populatiemodellering biedt integratie een systematische methode om geaccumuleerde grootheden te bepalen op basis van bekende veranderingssnelheden. Een toepassing hiervan is te vinden in de ecologie, waar het totale gewicht van een vispopulatie in een water wordt aangeduid als de biomassa. Wanneer de groeisnelheid van deze biomassa bekend is als functie van de tijd, kan met behulp van de analyse de totale biomassa op een toekomstig tijdstip worden bepaald.
Groeisnelheid en biomassafunctie
De groeisnelheid van de vispopulatie wordt voorgesteld door een functie G(t), waarbij t de tijd in jaren voorstelt en G(t) de veranderingssnelheid van de biomassa in kilogrammen per jaar weergeeft. De biomassa zelf wordt beschreven door een functie B(t). Aangezien de groeisnelheid aangeeft hoe de biomassa in de loop van de tijd verandert, wordt de biomassafunctie verkregen door de groeisnelheidsfunctie ten opzichte van de tijd te integreren.
Gebruik van de substitutiemethode
Om de integraal die de biomassafunctie definieert te berekenen, wordt de substitutiemethode toegepast. Deze methode houdt in dat een geschikte substitutievariabele u en de bijbehorende differentiaal du worden gekozen om de integraand te vereenvoudigen. Na het uitvoeren van de substitutie wordt de integraal herschreven in termen van u, waardoor deze eenvoudiger te evalueren is. Zodra de integratie is voltooid, wordt de verkregen uitdrukking teruggeschreven in de oorspronkelijke variabele t, wat resulteert in een algemene vorm van de biomassafunctie die een onbekende integratieconstante bevat.
Toepassen van beginvoorwaarden
De beginbiomassa van de vispopulatie in het jaar 2000 is gegeven en fungeert als beginvoorwaarde. Door de bijbehorende tijdwaarde en biomassa in de algemene biomassafunctie in te vullen, kan de integratieconstante worden bepaald. Deze stap zorgt ervoor dat het model de bekende toestand van het systeem op het begintijdstip correct weergeeft. Na het vaststellen van deze constante wordt de biomassafunctie herschreven in de definitieve vorm.
Bepalen van de biomassa in 2005
Om de biomassa van de vispopulatie in het jaar 2005 te bepalen, wordt de tijdsvariabele gelijkgesteld aan vijf jaar na 2000. Door deze waarde in de definitieve biomassafunctie in te vullen en de uitdrukking te vereenvoudigen, wordt de totale biomassa van de vispopulatie in 2005 verkregen.

Groeimodellen met integratie: probleemoplossing

In de populatiemodellering biedt integratie een systematische methode om geaccumuleerde grootheden te bepalen op basis van bekende veranderingssnelheden. ...

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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