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

在水体中，鱼类种群的总重量被称为鱼类生物量。鱼类种群的增长速率可以用函数 G(t) 来建模，其中 t 为年数，G(t) 为以千克/年为单位测量的增长速率。2000年的初始生物量已给出。目标是找出 2005 年鱼类种群的生物量。为此，需对速率函数进行积分，以求得生物量函数 B(t)。代入法通过选取 u 及其导数 du 来求解积分。对 u 的积分结果进行简化，然后将表达式还原，得到 B(t) 其中 c 为未知常数。已知量作为初始条件。代入这些值并求解后，可求得未知常数 c。然后将此 c 代入方程，重写 B(t)。为了求出2005年的生物量，将 t 等于 5 代入该生物量方程中。简化方程后，得到了生物量函数，该函数显示了2005年鱼类种群的总生物量。

利用积分建立增长模型：问题求解

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growth-models-with-integration-problem-solving

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Integrals

积分

KEY TERMS AND CONCEPTS

关键术语与概念

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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