Question

If \(\iint_{R} f(x, y) d A=\iint_{R} g(x, y) d A,\) does this imply \(f(x, y)=\) \(g(x, y) ?\)

Short answer

No, it does not imply \( f(x, y) = g(x, y) \). The functions can have the same integral over a region but differ pointwise.

Step-by-step solution

Understand the Problem

We are given two double integrals over the same region \( R \) with the same value: \( \iint_{R} f(x, y) \, dA = \iint_{R} g(x, y) \, dA \). The question asks if this implies that the functions \( f(x, y) \) and \( g(x, y) \) are equal everywhere on \( R \).

Consider the Integral Property

The property of integrals states that if two integrals over the same region are equal, it means the accumulated values of the functions over that region are identical. However, this does not necessarily mean that the functions themselves are equal pointwise.

Concepts of Function Equality

For two functions to be equal, \( f(x, y) = g(x, y) \), they must have the same value at every point \((x, y)\) in their domain. The equality of integrals only indicates that their total accumulation over \( R \) is the same, not their pointwise values.

Counterexample Investigation

Consider disjoint function behaviors. For example, let \( f(x, y) = 1 \) for \( x < 0 \) and 0 for \( x \geq 0 \), and \( g(x, y) = 0 \) for \( x < 0 \) and 1 for \( x \geq 0 \). Adjust the region or functions so their integrals are the same, although they differ pointwise, illustrating they can have the same integral without being identical.

Conclusion

Based on the properties of integrals and the exploration of possible function behaviors, the equality of double integrals over a region does not imply that the functions are necessarily equal pointwise across that region.

Key concepts

Function Equality

Function equality is all about whether two functions produce the same output for every single input in their domain. In simpler terms, if we say that two functions, say \( f(x, y) \) and \( g(x, y) \), are equal, we are really saying that for every coordinate point \((x, y)\) where they both exist, they give back the same result. However, recognizing function equality isn't always straightforward, especially in higher dimensions or more complex functions. Just because two formulas might look different doesn't mean they output differently at every point. Conversely, two different expressions can indeed have the same output across a domain.The takeaway here is that checking if functions are equal means verifying their pointwise values. This means evaluating both functions at every possible \((x, y)\) in their domain. Only then can we say with certainty whether they're the same function.

Integral Properties

Double integrals have some unique and useful properties that help in analyzing functions across two-dimensional regions. One key property is that they measure the total accumulation of a function across a region in the plane.

• If two double integrals over the same region are equal, this does not necessarily imply the integrands, the functions being integrated, are equal at every point. It simply means their total contributions (like "weights" if you imagine them as surfaces) to that integral over the region are the same.
• This scenario can be likened to two different paths that lead to the same destination. The journey or "path," in this case, the pointwise function values, could be different, but the total distance—the integral—is the same.

Understanding this conceptual difference is critical when working with integrals, as it highlights that an equal integral value is not always indicative of function equality.

Region of Integration

The region of integration, often denoted as \( R \), is a crucial component of the integration process, particularly in double integrals. Think of it as the specific "gathering space" over the plane where the function is being studied. When performing a double integral like \( \iint_{R} f(x, y) \, dA \), the region \( R \) defines the boundaries where \( f(x, y) \) is evaluated and accumulated. This region could be as straightforward as a rectangle or circle or as complex as a free-form shape defined by inequalities. Understanding \( R \) isn't just about knowing the shape but also the limits of integration. These limits are the bounds for \( x \) and \( y \) during integration and can significantly affect the integral's result. Without the same region \( R \), comparing two function integrals wouldn't be valid, as each might capture different parts of the plane.In practice, always ensure that the region of integration is clearly defined and consistent when evaluating or comparing double integrals.