In a large city, the amount of sulfur dioxide pollutant...

In a large city, the amount of sulfur dioxide pollutant released into the atmosphere due to the burning of coal and oil for heating purposes is given approximately by $$ P(n)=1+\cos \frac{\pi n}{26} \quad 0 \leq n \leq 104 $$ where $P(n)$ is the amount of sulfur dioxide (in tons) released during the $n$ th week after January $1 .$ (A) How many tons of pollutants were emitted into the atmosphere over the 2 -year period? (B) How many tons of pollutants were emitted into the atmosphere from $n=13$ to $n=52 ?$ (C) Illustrate part (B) graphically with an appropriate shaded region representing the total tons of pollutants emitted into the atmosphere.

In a large city, the amount of sulfur dioxide pollutant released into the atmosphere due to the burning of coal and oil for heating purposes is given approximately by
$$
P(n)=1+\cos \frac{\pi n}{26} \quad 0 \leq n \leq 104
$$
where $P(n)$ is the amount of sulfur dioxide (in tons) released during the $n$ th week after January $1 .$
(A) How many tons of pollutants were emitted into the atmosphere over the 2 -year period?
(B) How many tons of pollutants were emitted into the atmosphere from $n=13$ to $n=52 ?$
(C) Illustrate part (B) graphically with an appropriate shaded region representing the total tons of pollutants emitted into the atmosphere.

Key Concepts

Definite Integral as Area Under the Curve

The definite integral represents the accumulation of quantities, such as pollutant emissions, over an interval by calculating the area under a curve. Even though the problem deals with discrete weeks, the idea of integrating a continuous function to find total accumulation is conceptually similar to summing contributions over time. The shaded area in a graph visually represents this accumulation.

Graphical Interpretation and Shaded Regions

Graphical interpretation with shaded regions is used to visually convey the concept of accumulation or total quantity. In problems like this, shading the region under the curve for a given interval highlights the areas representing the total pollutant emissions. This visual tool aids in understanding both the behavior of the function and the concept of integration or summation over an interval.

Periodic Functions

Periodic functions repeat their values in regular intervals or periods. In the context of the problem, the cosine function has a specific period determined by the constant multiplying the variable, which is critical for understanding the behavior of the pollutant emission over time. Analyzing periodicity helps in identifying patterns, simplifications, or symmetries that can be leveraged in calculations.

Discrete Summation

Discrete summation involves adding a sequence of values that are determined by a function at specific, distinct points, such as weeks. In this problem, pollutant emissions are provided for each week, and the total emission over a period is calculated by summing these weekly amounts. This concept is fundamental in scenarios where data or functions are defined on discrete intervals.

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