APPLY 11 sales Sales of snowblowers are seasonal. Suppose...

APPLY 11 sales Sales of snowblowers are seasonal. Suppose the sales of snowblowers in one region of the country are approximated by $S(t)=500+500 \cos \left(\frac{\pi}{6} t\right)$ where $t$ is time (in months), with $t=0$ corresponding to November. The figure below shows a graph of $S .$ Use a definite integral to find total sales over a year.

APPLY 11 sales Sales of snowblowers are seasonal.
Suppose the sales of snowblowers in one region of the country
are approximated by

$S(t)=500+500 \cos \left(\frac{\pi}{6} t\right)$

where $t$ is time (in months), with $t=0$ corresponding to
November. The figure below shows a graph of $S .$ Use a definite
integral to find total sales over a year.

Key Concepts

Definite Integral

A definite integral is used to compute the total or accumulated change of a quantity over a specified interval. It represents the net area under the curve of the function, which in many applied contexts (like sales over time) corresponds to the total quantity accumulated over that period.

Accumulation of a Rate Function

When a function describes a rate of change, such as monthly sales, its definite integral over a specified time period gives the total value accumulated during that interval. This is a fundamental application of integration in scenarios where the instantaneous rate is known and the cumulative effect is desired.

Integration of Trigonometric Functions

Many models involving periodic behaviors, such as seasonal variations, include trigonometric functions (e.g., cosine). The integration of these functions involves standard techniques for handling trigonometric antiderivatives, which are essential for accurately evaluating accumulated quantities over an interval.

Periodic and Seasonal Modeling

Periodic functions are used in mathematical modeling to represent systems or processes that repeat at regular intervals, such as seasonal sales. Understanding the properties and integration of these functions is crucial for analyzing and predicting the behavior of systems that exhibit cyclic patterns.

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