نماذج النمو باستخدام التكامل: حل المسائل

Biomass is the total weight of a fish population in a water body. The growth rate of this biomass is modeled by a function G(t), measured in kilograms per year, where t represents time in years. By integrating the growth rate function, we obtain the biomass function B(t), which describes how total population weight changes over time.

To find the biomass function B(t) from a growth rate function G(t), you integrate the growth rate with respect to time. The substitution rule applied to indefinite integrals is commonly used to evaluate such integrals. Integration produces a general form of B(t) that includes an unknown constant of integration.

Initial conditions, such as the known biomass in the year 2000, allow you to find the constant of integration in the general biomass function. By substituting the initial time value and corresponding biomass into the equation, you solve for the unknown constant. This ensures the model accurately reflects the actual state of the fish population at the starting time.

The substitution method simplifies complex integrals by introducing a new variable u and its differential du. This transformation makes the integrand easier to evaluate. After integration, the expression is converted back to the original variable t, yielding a more manageable form of the biomass function.

Once the biomass function B(t) is determined with the constant of integration found, substitute the desired time value into the equation. For example, to find biomass in 2005 (five years after 2000), substitute t equals 5 into B(t) and simplify to obtain the total biomass at that year.

Growth rate G(t) describes how biomass changes per unit time, while biomass B(t) represents the accumulated total weight. Since biomass is the integral of growth rate, the fundamental theorem of calculus connects these functions: B'(t) equals G(t), meaning the derivative of biomass equals the growth rate.

Integration transforms known rates of change into accumulated quantities. In fish population ecology, integrating the growth rate function yields the biomass function, enabling predictions of future population weights. This application of integration problem solving demonstrates how calculus models ecological systems and supports conservation planning.