Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Colorado is a rectangular state (if we ignore the curvature of the earth). Let \(f(x, y)\) be the number of inches of rainfall during 2005 at the point \((x, y)\) in that state. What does \(f(x, y) d A\) represent? What does this number divided by Colorado the area of Colorado represent?

Step-by-Step Solution

Answer

The integral \(f(x, y) dA\) represents total rainfall; divided by area gives average rainfall.

Step 1: Understanding the Expression

The expression \(f(x, y) dA\) involves the function \(f(x, y)\), which represents rainfall in inches at a specific location \((x, y)\). The term \(dA\) represents an infinitesimally small area at the point \((x, y)\) in Colorado.

Step 2: Interpreting the Integral

The integral \(\int \int f(x, y) \, dA\) over the area of Colorado calculates the total rainfall in inches across the entire state for the year 2005. This is because it sums up the rainfall for each infinitesimal area \(dA\) by evaluating \(f(x, y)\) at each point \((x, y)\).

Step 3: Calculating Average Rainfall

Dividing the total rainfall (found by evaluating the integral) by the total area of Colorado gives the average rainfall per square unit across the state. Specifically, this is computed as \(\frac{1}{\text{area of Colorado}} \int \int f(x, y) \, dA\).

Key Concepts

Average Rainfall Calculation

Imagine you're trying to determine how much rain fell, on average, over an entire state like Colorado in the year 2005. Rain doesn't fall evenly everywhere. Some spots may get more rain and others less. To account for this variability, mathematicians use multiple integrals. The average rainfall can be defined as the total rainfall divided by the area over which it fell. This gives us an idea of how much rain, on average, can be expected per unit area.

To put this into a mathematical perspective, you'd first calculate the total rainfall using the double integral\[ \int \int f(x, y) \, dA \]where \(f(x, y)\) is the function representing rainfall at a specific location \((x, y)\), and \(dA\) is the small area about \((x, y)\). Once you have this total rainfall value, you'd then find the average by dividing by Colorado's total area:\[\frac{1}{\text{area of Colorado}} \int \int f(x, y) \, dA.\]

This approach efficiently provides an average value by summing up all the rain that fell in every tiny part and dividing it evenly across the whole region.

Rainfall Distribution Function

The function \(f(x, y)\) plays a crucial role in understanding how rain varied across the state of Colorado. Known as the rainfall distribution function, this mathematical expression details how much rain fell at each point \((x, y)\) in inches. By examining \(f(x, y)\), one can determine where the rainfall was highest and lowest, providing a detailed picture of the geographical rainfall patterns.

Given the dynamic nature of weather, this function allows us to quantify differences in local climate within the state. This can be visualized as a 3D surface over Colorado, where the height of the surface at each point corresponds to the rainfall received there. Areas with higher values might correlate with mountain ranges or regions that experienced more precipitation during the year.

• \(f(x, y)\) helps in pinpointing rain-heavy areas.
• It provides insights into local agricultural planning.
• Understanding rainfall variability can aid disaster management efforts.

Thus, the rainfall distribution function is more than just numbers; it is a critical tool for decision-making and analysis.

Integration over a Region

At the core of calculating total rainfall or average rainfall is the mathematical operation known as integration over a region. This is a fundamental concept in multivariable calculus, enabling the calculation of a total quantity spread across a defined space. When we talk about integration over a region in the context of rainfall, it means adding up all the small contributions of rainfall from every tiny part of Colorado to find the total.

This involves using the double integral\[ \int \int f(x, y) \, dA \]which efficiently aggregates data from a continuous distribution across geographic coordinates. Every point \((x, y)\) contributes a small amount \(f(x, y) \, dA\) to the sum, reflecting its particular rainfall.

Applying this concept means:

• Tackling real-world problems that involve spatial data, like environmental impact studies.
• Utilizing calculus to encompass an entire geographic region within mathematical models.
• Understanding the cumulative impact of distributed quantities, such as rain, which can vary significantly over a large area.

This understanding is instrumental in not only mathematics but also in fields like meteorology and environmental science, maintaining a bridge between theoretical mathematics and practical applications.