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

水域において、魚の総重量は魚のバイオマスとして知られています。魚類の個体群の成長率は関数G(t)でモデル化でき、ここでtは年数、G(t)は年間キログラム単位で測定される成長率です。2000年の初期バイオマスが示されています。目標は2005年の魚類個体数のバイオマスを特定することです。これを求めるために、レート関数を積分してバイオマス関数B(t)を求めます。置換法は、Uとその微分duを取ることで積分を解くために用いられます。得られるuへの積分は簡略化され、式を書き戻して未知の定数cを持つB(t)を得る。既知の量を初期条件として取る。これらの値を代入して解くことで、未知の定数 c が見つかります。このcを式に代入すると、B(t)が書き換えられます。2005年のバイオマスを求めるために、このバイオマス方程式にtを5に代入します。方程式を簡略化すると、バイオマス関数が導かれ、2005年時点での魚類個体群の総バイオマスが示されます。

積分を用いた成長モデル：問題解決

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growth-models-with-integration-problem-solving

Education

JoVE Core

Calculus

微積分学

Unit

単位

Integrals

積分

KEY TERMS AND CONCEPTS

主要用語と概念

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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