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

Bir su kütlesinde, balık popülasyonunun toplam ağırlığı balık biyokütlesi olarak bilinir. Bir balık popülasyonunun büyüme hızı, t yıl sayısı, G(t) ise yıllık kilogram cinsinden ölçülen büyüme oranı olan G(t) fonksiyonuyla modellenebilir.2000 yılı için ilk biyokütle verilmiştir.Amaç, 2005 yılında balık popülasyonunun biyokütlesini bulmaktır.Bunu bulmak için, hız fonksiyonu entegre edilip biyokütle fonksiyonu B(t)'yi bulur.İntemalı yöntemi, integrali çözmek için u ve türevi du alınır.U'daki ortaya çıkan integral basitleştirilir ve ifade bilinmeyen bir sabit c ile B(t) elde etmek için geri yazılır.Bilinen miktarlar başlangıç koşulları olarak alınır. Bu değerler yerine koyup çözüldükten sonra, bilinmeyen sabit c bulunur. Daha sonra, bu c denklemine koyulduğunda, B(t) yeniden yazılır.2005 yılında biyokütle bulmak için, t 5'e eşit bu biyokütle denklemine konur.Denklemi basitleştirdikten sonra, 2005 yılında balık popülasyonunun toplam biyokütlesi gösterilen biyokütle fonksiyonu bulunmuştur.

İntegralle büyüme modelleri: problem Çözme

N&#xFC;fus modellemesinde integral alma, bilinen de&#x11F;i&#x15F;im h&#x131;zlar&#x131;ndan birikimli b&#xFC;y&#xFC;kl&#xFC;kleri belirlemek i&#xE7;in sistematik bir y&#xF6;ntem sunar. Bunun bir uygulamas&#x131; ekolojide g&#xF6;r&#xFC;l&#xFC;r: Bir su ortam&#x131;ndaki bal&#x131;k pop&#xFC;lasyonunun toplam a&#x11F;&#x131;rl&#x131;&#x11F;&#x131;na biyok&#xFC;tle denir. Bu biyok&#xFC;tlenin b&#xFC;y&#xFC;me h&#x131;z&#x131; zaman&#x131;n bir fonksiyonu olarak biliniyorsa, gelecekteki bir tarihte toplam biyok&#xFC;tleyi belirlemek i&#xE7;in kalk&#xFC;l&#xFC;s kullan&#x131;labilir.
B&#xFC;y&#xFC;me Oran&#x131; ve Biyok&#xFC;tle Fonksiyonu
Bal&#x131;k pop&#xFC;lasyonunun b&#xFC;y&#xFC;me h&#x131;z&#x131; G(t) fonksiyonu ile g&#xF6;sterilsin; burada t y&#x131;l cinsinden zaman&#x131;, G(t) ise biyok&#xFC;tlenin kilogram/y&#x131;l cinsinden de&#x11F;i&#x15F;im h&#x131;z&#x131;n&#x131; temsil eder. Biyok&#xFC;tle B(t) fonksiyonu ile g&#xF6;sterilir. B&#xFC;y&#xFC;me h&#x131;z&#x131;, biyok&#xFC;tlenin zamana ba&#x11F;l&#x131; de&#x11F;i&#x15F;imini tan&#x131;mlad&#x131;&#x11F;&#x131;ndan, biyok&#xFC;tle fonksiyonu, b&#xFC;y&#xFC;me h&#x131;z&#x131; fonksiyonunun zamana g&#xF6;re integrallenerek elde edilir.
De&#x11F;i&#x15F;ken D&#xF6;n&#xFC;&#x15F;&#xFC;m&#xFC; (Yerine Koyma) Y&#xF6;nteminin Kullan&#x131;m&#x131;
Biyok&#xFC;tle fonksiyonunu tan&#x131;mlayan integrali hesaplamak i&#xE7;in de&#x11F;i&#x15F;ken d&#xF6;n&#xFC;&#x15F;&#xFC;m&#xFC; (yerine koyma) y&#xF6;ntemi kullan&#x131;l&#x131;r. Bu y&#xF6;ntem, integrali basitle&#x15F;tirmek amac&#x131;yla uygun bir u de&#x11F;i&#x15F;keni ve onun diferansiyeli du&#x2019;nun se&#xE7;ilmesini i&#xE7;erir. Yerine koyma yap&#x131;ld&#x131;ktan sonra integral u cinsinden yeniden yaz&#x131;l&#x131;r; b&#xF6;ylece integralin hesaplanmas&#x131; daha kolay h&#xE2;le gelir. &#x130;ntegral alma i&#x15F;lemi tamamland&#x131;ktan sonra ifade yeniden t de&#x11F;i&#x15F;kenine d&#xF6;n&#xFC;&#x15F;t&#xFC;r&#xFC;l&#xFC;r ve bilinmeyen bir integrasyon sabiti (integral sabiti) i&#xE7;eren biyok&#xFC;tle fonksiyonunun genel bi&#xE7;imi elde edilir.
Ba&#x15F;lang&#x131;&#xE7; Ko&#x15F;ulunun Uygulanmas&#x131;
2000 y&#x131;l&#x131;ndaki bal&#x131;k pop&#xFC;lasyonunun ba&#x15F;lang&#x131;&#xE7; biyok&#xFC;tlesi verilmi&#x15F;tir ve bu de&#x11F;er bir ba&#x15F;lang&#x131;&#xE7; ko&#x15F;ulu olarak kullan&#x131;l&#x131;r. &#x130;lgili zaman de&#x11F;eri ve biyok&#xFC;tle, genel biyok&#xFC;tle fonksiyonunda yerine yaz&#x131;larak integrasyon sabiti belirlenir. Bu ad&#x131;m, modelin ba&#x15F;lang&#x131;&#xE7; an&#x131;nda sistemin bilinen durumunu do&#x11F;ru bi&#xE7;imde yans&#x131;tmas&#x131;n&#x131; sa&#x11F;lar. Sabit belirlendikten sonra biyok&#xFC;tle fonksiyonu son bi&#xE7;imiyle yeniden yaz&#x131;l&#x131;r.
2005 Y&#x131;l&#x131;ndaki Biyok&#xFC;tlenin Belirlenmesi
2005 y&#x131;l&#x131;ndaki bal&#x131;k pop&#xFC;lasyonunun biyok&#xFC;tlesini bulmak i&#xE7;in zaman de&#x11F;i&#x15F;keni 2000 y&#x131;l&#x131;ndan be&#x15F; y&#x131;l sonras&#x131;na e&#x15F;itlenir. Bu de&#x11F;er nihai biyok&#xFC;tle fonksiyonunda yerine yaz&#x131;l&#x131;p sadele&#x15F;tirildi&#x11F;inde 2005 y&#x131;l&#x131;ndaki bal&#x131;k pop&#xFC;lasyonunun toplam biyok&#xFC;tlesi elde edilir.

İntegralle Büyüme Modelleri: Problem Çözme

N&#xFC;fus modellemesinde integral alma, bilinen de&#x11F;i&#x15F;im h&#x131;zlar&#x131;ndan birikimli b&#xFC;y&#xFC;kl&#xFC;kleri belirlemek i&#xE7;in ...

growth-models-with-integration-problem-solving

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Integrals

İntegraller

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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