Problem 12 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. If \(\sigma\) is the portion of a plane \(z=c\) over a region \(R\) in the \(x y\) -plane, then $$ \iint_{\sigma} f(x, y, z) d S=\iint_{R} f(x, y, c) d A $$ for every continuous function \(f\) on \(\sigma .\)

Step-by-Step Solution

Answer

The statement is true; the integrals are equal when the surface is a flat plane.

Step 1: Understand the Given Problem

We are given a surface integral on the portion of the plane \( z = c \) over a region \( R \) in the \( xy \)-plane, denoted as \( \sigma \). We need to determine if the surface integral over \( \sigma \) of a continuous function \( f(x, y, z) \) is equal to the double integral of \( f(x, y, c) \) over the region \( R \).

Step 2: Interpret the Surface Integral Definition

The surface integral \( \iint_{\sigma} f(x, y, z) \, dS \) represents the integral of \( f \) over the surface defined by \( \sigma \). The differential element \( dS \) is the area element on the surface, which depends on the orientation of the surface.

Step 3: Recognize the Plane Condition

The surface \( \sigma \) is the plane \( z = c \). In this case, the surface is flat and parallel to the \( xy \)-plane, which means the normal vector is constant, specifically \( (0, 0, 1) \). This results in the differential area element \( dS = dA \) because the surface is horizontal.

Step 4: Evaluate the Integral

Given the condition \( dS = dA \) as the surface is flat and parallel to the \( xy \)-plane, the surface integral becomes \( \iint_{\sigma} f(x, y, z) \, dS = \iint_{R} f(x, y, c) \, dA \). In this case, the function \( f(x, y, z) \) evaluated at the plane \( z = c \) simplifies to \( f(x, y, c) \), making the statement true.

Step 5: Conclude the Evaluation

For the given plane \( z = c \), since \( dS = dA \) on this surface and \( f(x, y, z) \) can be rewritten as \( f(x, y, c) \) in terms of \( R \), the original equation holds as both integrals are indeed equal.

Key Concepts

Understanding the Plane: Flat and Consistent

A plane is a flat, two-dimensional surface that extends infinitely in all directions. In mathematics, a plane can be described with a linear equation like \( z = c \). In this specific equation, \( c \) is a constant value which shows how high or low the plane is from the \( xy \)-axis at a constant height along the \( z \)-axis.
When you look at the plane \( z = c \), it lies flat and parallel to the \( xy \)-coordinate plane because the value of \( z \) never changes. This plane is exactly like a sheet lying perfectly flat on the floor.

• Because it's flat and consistent, analyzing any function across this surface becomes straightforward as it maintains the same orientation.
• This means that any calculation involving area is simply like calculating areas on a flat \( xy \)-plane, especially when considering a region \( R \) on this plane.

Continuous Function Simplified

A continuous function is one where small changes in the input lead to small changes in the output. This is a key concept in mathematics because it ensures that there are no jumps or gaps in the graph of the function. In other words, you can draw the graph of a continuous function without lifting your pencil.
For the function \( f(x, y, z) \) on our plane \( z = c \), it simplifies into \( f(x, y, c) \) since the \( z \)-coordinate is constant. Here's how continuous functions make life easier on this plane:

• No unexpected surprises: Knowing \( f \) is continuous means the function will behave predictably over the region \( R \).
• Simplification of evaluation: Instead of worrying about three variables \( x, y, z \), the plane's consistent height allows focusing only changes in \( x \) and \( y \).
• Continuity ensures that the math works as expected without encountering undefined values.

Double Integral on the Plane

Double integrals are a way to add up values over a two-dimensional region, often used to find areas, volumes, and other significant quantities. When working with double integrals over a region \( R \) in the \( xy \)-plane, we use them to integrate functions of two variables: \( f(x, y) \).
Since the plane \( z = c \) is flat and parallel to the \( xy \)-plane, the double integral \( \iint_{R} f(x, y, c) \, dA \) represents the sum of values of \( f(x, y, c) \) over this region.

• It "projects" the function from the plane to the \( xy \)-plane, making calculations simple and straightforward.
• The integral handles the addition of values within the bounded area \( R \).
• Since \( dS = dA \) on this flat plane, evaluating \( f(x, y, c) \, dA \) doesn't require complicated calculations or transformations.

Thus, double integrals over a region in a plane simplify many problems from physics to engineering, where surface areas and quantities need to be evaluated.