What is the result of integrating a population growth rate between times \( t=a \) and \( t=b \), where \( b>a \) ? Choose the correct answer. A. Integrating a population growth rate results in the rate of change of the population from time a to time \( b \). B. Integrating a population growth rate results in the net change in the population growth rate from time a to time \( b \). C. Integrating a population growth rate results in the net change of the population from time a to time \( b \).

To determine the result of integrating a population growth rate between times \( t = a \) and \( t = b \), let's break down what integration represents in this context. **Population Growth Rate (\( r(t) \))**: - The population growth rate \( r(t) \) is the rate at which the population is changing at any given time \( t \). Mathematically, this can be expressed as: \[ r(t) = \frac{dP(t)}{dt} \] where \( P(t) \) is the population at time \( t \). **Integration of the Growth Rate**: - Integrating the growth rate \( r(t) \) from \( t = a \) to \( t = b \) gives: \[ \int_{a}^{b} r(t) \, dt = \int_{a}^{b} \frac{dP(t)}{dt} \, dt = P(b) - P(a) \] - This integral represents the **net change in the population** between times \( a \) and \( b \). **Evaluating the Options**: - **Option A**: Refers to the rate of change of the population, which is actually \( r(t) \), not the integral of \( r(t) \). - **Option B**: Discusses the net change in the population growth rate itself, which is not represented by the integral. - **Option C**: Accurately describes the result of the integral as the net change in the population from time \( a \) to time \( b \). **Correct Answer**: **C. Integrating a population growth rate results in the net change of the population from time a to time \( b \).**