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

W zbiorniku wodnym całkowita masa populacji ryb nazywana jest biomasą ryb. Tempo wzrostu populacji ryb można modelować za pomocą funkcji G(t), gdzie t to liczba lat, a G(t) to tempo wzrostu mierzone w kilogramach rocznie.Podano początkową biomasę za rok 2000.Celem jest ustalenie biomasy populacji ryb w roku 2005.Aby to znaleźć, całkuje się funkcję szybkości, aby znaleźć funkcję biomasy B(t).Metoda podstawienia służy do rozwiązania całki poprzez wzięcie u i jej pochodnej du.Powstała całka w u jest upraszczana, a wyrażenie zapisywane jest z powrotem, aby uzyskać B(t) o nieznanej stałej c.Znane wielkości są traktowane jako warunki początkowe. Po podstawieniu tych wartości i rozwiązaniu otrzymuje się nieznaną stałą c. Następnie, podstawiając to c do równania, B(t) jest przepisywane.Aby wyznaczyć biomasę w roku 2005, t równa się 5 jest podstawione do tego równania biomasy.Po uproszczeniu równania znajduje się funkcję biomasy, która pokazuje całkowitą biomasę populacji ryb w 2005 roku.

modele wzrostu z wykorzystaniem całkowania: rozwiązywanie problemów

W modelowaniu populacji ca&#x142;kowanie zapewnia systematyczny spos&#xF3;b okre&#x15B;lania skumulowanych ilo&#x15B;ci na podstawie znanych szybko&#x15B;ci zmian. Jedno z takich zastosowa&#x144; pojawia si&#x119; w ekologii, gdzie ca&#x142;kowita masa ryb w zbiorniku wodnym jest okre&#x15B;lana jako biomasa. Znaj&#x105;c szybko&#x15B;&#x107; wzrostu tej biomasy jako funkcj&#x119; czasu, mo&#x17C;na wykorzysta&#x107; rachunek ca&#x142;kowy do okre&#x15B;lenia ca&#x142;kowitej biomasy w przysz&#x142;o&#x15B;ci.
Tempo wzrostu i funkcja biomasy
Niech tempo wzrostu populacji ryb b&#x119;dzie reprezentowane przez funkcj&#x119; G(t), gdzie t oznacza czas mierzony w latach, a G(t) reprezentuje szybko&#x15B;&#x107; zmiany biomasy w kilogramach na rok. Sama biomasa jest reprezentowana przez funkcj&#x119; B(t). Poniewa&#x17C; tempo wzrostu opisuje, jak biomasa zmienia si&#x119; w czasie, funkcj&#x119; biomasy uzyskuje si&#x119; poprzez ca&#x142;kowanie funkcji tempa wzrostu wzgl&#x119;dem czasu.
Zastosowanie metody podstawiania
Aby wyznaczy&#x107; ca&#x142;k&#x119; definiuj&#x105;c&#x105; funkcj&#x119; biomasy, stosuje si&#x119; metod&#x119; podstawiania. Polega ona na wybraniu odpowiedniej zmiennej podstawiania u oraz jej r&#xF3;&#x17C;niczki du w celu uproszczenia ca&#x142;ki. Po wykonaniu podstawiania ca&#x142;ka jest zapisywana w zale&#x17C;no&#x15B;ci od zmiennej u, co u&#x142;atwia jej obliczenie. Po zako&#x144;czeniu ca&#x142;kowania wyra&#x17C;enie jest przekszta&#x142;cane z powrotem do pierwotnej zmiennej t, co prowadzi do og&#xF3;lnej postaci funkcji biomasy, kt&#xF3;ra zawiera nieznan&#x105; sta&#x142;&#x105; ca&#x142;kowania.
Zastosowanie warunk&#xF3;w pocz&#x105;tkowych
Podana jest pocz&#x105;tkowa biomasa populacji ryb w roku 2000, kt&#xF3;ra stanowi warunek pocz&#x105;tkowy. Podstawiaj&#x105;c odpowiedni&#x105; warto&#x15B;&#x107; czasow&#x105; i biomas&#x119; do og&#xF3;lnej funkcji biomasy, mo&#x17C;na wyznaczy&#x107; sta&#x142;&#x105; ca&#x142;kowania. Ten krok zapewnia, &#x17C;e model dok&#x142;adnie odzwierciedla znany stan uk&#x142;adu w momencie pocz&#x105;tkowym. Po wyznaczeniu tej sta&#x142;ej funkcja biomasy jest zapisywana w postaci ostatecznej.
Wyznaczanie biomasy w 2005 roku
Aby okre&#x15B;li&#x107; biomas&#x119; populacji ryb w 2005 roku, zmienn&#x105; czasow&#x105; ustawia si&#x119; na pi&#x119;&#x107; lat po 2000 roku. Podstawienie tej warto&#x15B;ci do sfinalizowanej funkcji biomasy i uproszczenie daje ca&#x142;kowit&#x105; biomas&#x119; populacji ryb w 2005 roku.

Modele wzrostu z wykorzystaniem całkowania: Rozwiązywanie problemów

W modelowaniu populacji ca&#x142;kowanie zapewnia systematyczny spos&#xF3;b okre&#x15B;lania skumulowanych ilo&#x15B;ci na podstawie znanych ...

growth-models-with-integration-problem-solving

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Integrals

Całki

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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.

calculus

Finding Fish Biomass with Integrals

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