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

In un corpo idrico, il peso totale della popolazione ittica è noto come biomassa di pesci. Il tasso di crescita di una popolazione di pesci può essere modellato dalla funzione G(t), dove t è il numero di anni e G(t) è il tasso di crescita misurato in chilogrammi all'anno.Viene fornita la biomassa iniziale per l'anno 2000.L'obiettivo è trovare la biomassa della popolazione ittica nell'anno 2005.Per trovare ciò, la funzione di velocità viene integrata per trovare la funzione biomassa B(t).Il metodo della sostituzione viene utilizzato per risolvere l'integrale prendendo u e la sua derivata du.L'integrale risultante in u viene semplificato, e l'espressione viene poi scritta di nuovo per ottenere B(t) con una costante sconosciuta c.Le quantità note sono prese come condizioni iniziali. Dopo aver sostituito questi valori e risolto, si trova la costante sconosciuta c. Poi, sostituendo questa c nell'equazione, B(t) viene riscritto.Per trovare la biomassa nell'anno 2005, si sostituisce t uguale a 5 in questa equazione di biomassa.Dopo aver semplificato l'equazione, si trova la funzione biomassa, che mostra la biomassa totale della popolazione ittica nel 2005.

modelli di crescita con integrazione: risoluzione di problemi

Nella modellazione di popolazioni, l&#x2019;integrazione fornisce un metodo sistematico per determinare quantit&#xE0; accumulate a partire da tassi di variazione noti. Un&#x2019;applicazione tipica si riscontra in ecologia, dove il peso totale di una popolazione ittica in un corpo idrico &#xE8; detto biomassa. Quando il tasso di crescita di tale biomassa &#xE8; noto in funzione del tempo, il calcolo integrale consente di determinare la biomassa totale a una data futura.
Tasso di crescita e funzione di biomassa
Si rappresenti il tasso di crescita della popolazione ittica mediante una funzione G(t), dove t denota il tempo misurato in anni e G(t) rappresenta il tasso di variazione della biomassa in chilogrammi per anno. La biomassa stessa &#xE8; rappresentata da una funzione B(t). Poich&#xE9; il tasso di crescita descrive come la biomassa varia nel tempo, la funzione di biomassa si ottiene integrando la funzione del tasso di crescita rispetto al tempo.
Uso del metodo di sostituzione
Per valutare l&#x2019;integrale che definisce la funzione di biomassa, si impiega il metodo di sostituzione. Ci&#xF2; comporta la scelta di una variabile di sostituzione appropriata u e del relativo differenziale du per semplificare l&#x2019;integrando. Dopo aver effettuato la sostituzione, l&#x2019;integrale viene riscritto in termini di u, il che ne rende pi&#xF9; agevole la valutazione. Completata l&#x2019;integrazione, l&#x2019;espressione viene ricondotta alla variabile originale t, ottenendo una forma generale della funzione di biomassa che include una costante di integrazione incognita.
Applicazione delle condizioni iniziali
La biomassa iniziale della popolazione ittica nell&#x2019;anno 2000 &#xE8; data e costituisce una condizione iniziale. Sostituendo nella funzione di biomassa generale il valore temporale corrispondente e la biomassa, &#xE8; possibile determinare la costante d&#x2019;integrazione. Questo passaggio garantisce che il modello rappresenti correttamente lo stato del sistema al tempo iniziale. Una volta determinata tale costante, la funzione di biomassa viene riscritta nella sua forma finale.
Determinazione della biomassa nel 2005
Per determinare la biomassa della popolazione ittica nell&#x2019;anno 2005, la variabile temporale si pone a cinque anni dopo il 2000. Sostituendo questo valore nella funzione di biomassa definitiva e semplificando, si ottiene la biomassa totale della popolazione ittica nel 2005.

Modelli di crescita con integrazione: risoluzione di problemi

Nella modellazione di popolazioni, l&#x2019;integrazione fornisce un metodo sistematico per determinare quantit&#xE0; accumulate a partire da tassi di ...

growth-models-with-integration-problem-solving

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Integrals

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TERMINI E CONCETTI CHIAVE

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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growth models with integration: problem solving

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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