Problem 30 - Thomas Calculus

Question

Suppose \(f ( x , y )\) is continuous and nonnegative over a region \(R\) in the plane with a defined area \(A ( R ) .\) If \(\iint _ { R } f ( x , y ) d A = 0 ,\) prove that \(f ( x , y ) = 0\) at every point \(( x , y ) \in R\) .

Step-by-Step Solution

Answer

The function must be zero everywhere in the region R.

Step 1: Understand the Given Information

We know that the function \(f(x, y)\) is continuous and nonnegative over the region \(R\). Additionally, the area of the region is \(A(R)\) and the double integral of \(f(x, y)\) over \(R\) is zero: \(\iint _ { R } f ( x , y ) \, dA = 0\).

Step 2: Interpret the Integral Condition

The equation \(\iint _ { R } f(x, y) \, dA = 0\) means that the accumulated value of \(f(x, y)\) over the region \(R\) is \(0\). Since \(f(x, y)\) is nonnegative, \(f(x, y)\) must be zero almost everywhere in \(R\) for this integral to yield zero.

Step 3: Use the Continuity of f(x,y)

Since \(f(x, y)\) is continuous on \(R\), if it were not zero at any point in \(R\), there would be some neighborhood where \(f(x, y)\) is positive, contradicting the integral result. Hence, by continuity, \(f(x, y)\) must be exactly zero at every point in \(R\).

Step 4: Conclude with the Argument of Continuity and Nonnegativity

Combine nonnegativity and continuity to argue that if \(f(x, y)\) were non-zero at even a single point, the contribution to the integral could not be zero. Therefore, \(f(x, y)\) must be zero at every point within \(R\) to satisfy the condition \(\iint _ { R } f(x, y) \, dA = 0\).

Key Concepts

Continuity

In mathematics, continuity of a function like \( f(x, y) \) over a region \( R \) in the plane is an important concept. A function is continuous if, roughly speaking, you can draw it without lifting your pen from the paper. For \( f(x, y) \), this means that small changes in \( x \) and \( y \) result in small changes in \( f(x, y) \).
This smooth behavior ensures that there are no sudden jumps or breaks in the values of \( f(x, y) \) within the region \( R \).
The importance of continuity here is that if a function is non-zero at even a single point within \( R \), continuity would imply that \( f(x, y) \) would also take on positive values in some surrounding neighborhood.
However, because the integral of \( f(x, y) \) over \( R \) is zero, no such neighborhood can exist without contradicting the condition \( \iint _ { R } f(x, y) \, dA = 0 \). Therefore, \( f(x, y) \) is zero everywhere in \( R \).

• Continuity requires no breaks or jumps in \( f(x, y) \).
• It ensures local neighborhoods of \( (x, y) \) behave predictably.
• Supports the integral condition by prohibiting positive values over any neighborhood.

Nonnegative Functions

Nonnegative functions are those that do not take any negative values; their outputs are zero or positive. For the function \( f(x, y) \) defined over region \( R \), being nonnegative means that \( f(x, y) \geq 0 \) for every point in \( R \).
This attribute is vital when computing integrals because the integral value represents the accumulated area under the function's surface over the region \( R \).
If the double integral \( \iint _ { R } f(x, y) \, dA = 0 \), then the total 'area' accumulated by \( f(x, y) \) over \( R \) is zero.
With \( f(x, y) \geq 0 \), it's only possible for this total to be zero if \( f(x, y) = 0 \) almost everywhere in \( R \), given any positive value, no matter how small, would contribute positively to the integral.

• Nonnegative functions satisfy \( f(x, y) \geq 0 \).
• Important for interpreting the zero integral condition, which suggests zero accumulation.
• Key in ensuring the function must be zero throughout \( R \).

Region in the Plane

A region in the plane, denoted as \( R \), is a specific area over which the function \( f(x, y) \) is examined. This region can be described in many ways, such as being bounded by curves or lines. In the context of double integrals, the choice of region \( R \) matters as it defines where the function is integrated over.
The exercise specifies that \( f(x, y) \) is both continuous and nonnegative over \( R \), and the double integral over this region is zero. Because of this, within region \( R \), we examine all points to ensure that the behavior of \( f(x, y) \) aligns with the conditions provided (i.e., nonnegative, zero integration result).
Despite any shape or complexity of \( R \), these conditions remain consistent throughout the entire region. Analyzing how \( f(x, y) \) behaves over \( R \) gives insight into its complete adherence to continuity and nonnegativity, helping us conclude that \( f(x, y) \) must be zero throughout \( R \).

• Region \( R \) defines the boundary for analysis of \( f(x, y) \).
• Dictates integration limits for the double integral.
• Integral and function characteristics assessed over entire region \( R \).