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 \(\iint_{\text {Colorado }} f(x, y) d A\) represent? What does this number divided by the area of Colorado represent?

Step-by-Step Solution

Answer

The integral represents total rainfall; dividing it by Colorado's area gives average rainfall.

Step 1: Understanding the Integral

The integral \( \iint_{\text {Colorado}} f(x, y) \, dA \) calculates the total amount of rainfall received across the entire area of Colorado during 2005. The function \( f(x, y) \) represents the rainfall at each point \((x, y)\), and by integrating this function across all points in Colorado, we accumulate the rainfall over the entire state.

Step 2: Calculating the Average Rainfall

Dividing the total rainfall, \( \iint_{\text{Colorado}} f(x, y) \, dA \), by the area of Colorado yields the average rainfall per unit area. This represents the mean rainfall that would have been experienced uniformly across the state if the rain had fallen evenly.

Key Concepts

Rainfall Integration

Rainfall integration is a technique used to calculate the total rainfall over a region by using a double integral. In our exercise, the double integral \( \iint_{\text{Colorado}} f(x, y) \, dA \) represents the total inches of rainfall that fell on Colorado in 2005.

Here's why this works:

• The function \( f(x, y) \) gives us the rainfall at any specific point \( (x, y) \) in the state.
• The double integral sums up these rainfall values over every point in the area of Colorado.
• The notation \( dA \) is a small area element, which helps us account for rainfall at every possible location within the state's boundaries.

By using integration, we seamlessly gather the infinitely many small patches of rainfall into a single calculable total, providing an effective method to quantify weather data over large geographical areas.

Average Rainfall Calculation

Average rainfall calculation provides insight into how much rainfall would be received uniformly across a given area. To find the average rainfall across Colorado, we divide the total rainfall by the state's area.

This process involves two main steps:

• First, we compute the total rainfall using the double integral \( \iint_{\text{Colorado}} f(x, y) \, dA \).
• Second, we divide this result by Colorado's area, which is the physical size of the state on a map, usually in square miles or kilometers.

This division gives us the average rainfall, simplifying varied rainfall data into a single understandable figure. It answers the question of what would happen if the total rainfall was distributed evenly across the entire state, providing a useful metric for comparisons and predictions.

Area of a Region in Integrals

Understanding the area of a region in integrals is crucial for converting complex spatial data into practical information. In the context of our exercise, once we identify Colorado as a rectangular area, we can easily integrate over it.

Here's how it applies:

• The region's area appears as part of the division in calculating average values (channeling back to when we divided by Colorado's area).
• Determining the region's size ensures we apply the correct scale to our results, so any calculations reflect a true average.

In integrals, especially with physical applications, knowing the area lets us properly weigh different points within a target region—critical for ensuring all parts contribute proportionately to the total outcome.