A model for the basal metabolism rate, in kcal/h, of a young...

A model for the basal metabolism rate, in kcal/h, of a young man is $ R(t) = 85 - 0.18 \cos(\pi t/12) $, where $ t $ is the time in hours measured from 5:00 AM. What is the total basal metabolism of this man, $ \displaystyle \int^{24}_0 R(t) \, dt $, over a 24-hour time period?

Key Concepts

Mathematical Modeling of Rates

This concept involves using functions to represent dynamic rates, such as metabolic rates, over time. Integration of these rate functions over an interval yields the total accumulated quantity, thereby linking mathematical models to real-world phenomena.

Trigonometric Integration

Trigonometric integration involves techniques specifically tailored for integrating functions involving trigonometric functions like cosine or sine. Recognizing the behavior of these functions, especially their periodic properties, can simplify the evaluation of integrals over intervals corresponding to full periods.

Definite Integration

Definite integration calculates the net accumulation or total change of a quantity over a specific interval. In this context, it is used to determine the total amount of basal metabolism over a 24?hour period by integrating the rate function over time.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, enabling the computation of a definite integral by finding an antiderivative of the function and evaluating it at the endpoints. It is a core concept in using integration to calculate accumulated quantities, such as total energy usage.

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