Question

Explain why if \(f(x, y)>0\) over a region \(R,\) then \(\iint_{R} f(x, y) d A>0\).

Short answer

If \(f(x, y) > 0\) in \(R\), each part contributes positively to the integral, resulting in \(\iint_{R} f(x, y) d A > 0\).

Step-by-step solution

Understand the Double Integral

The double integral \(\iint_{R} f(x, y) d A\) represents the accumulation of values of the function \(f(x, y)\) over the region \(R\). It can be thought of as a kind of "volume" under the surface defined by \(f(x, y)\) over \(R\) when \(f(x, y)\) represents a height above the \(xy\)-plane.

Positive Function Implication

If the function \(f(x, y)\) is greater than 0 for all \((x, y)\) in the region \(R\), it means that no part of the "surface" described by \(f(x, y)\) dips below the \(xy\)-plane within \(R\). Thus, the values being "accumulated" in the integral are all positive.

Accumulate Positive Contributions

Because \(f(x, y) > 0\) everywhere in \(R\), each infinitesimally small area of \(dA\) contributes a positive amount \(f(x, y)\,dA\) to the integral. Since each of these contributions is positive, the sum total, or the integral, cannot be zero or negative—it must be positive.

Conclusion from Positive Integral Contributions

Since the double integral \(\iint_{R} f(x, y) d A\) is the sum of all these positive values, it follows logically that \(\iint_{R} f(x, y) d A > 0\). This is because you are summing positive quantities over a region \(R\) with a non-zero area.

Key concepts

Positive Function

A positive function is one where the output, or value, is always greater than zero for every point in its domain. In the context of a double integral, the function we're focusing on assigns a `height` above the zero level of the xy-plane.
If the function \( f(x, y) \) is positive over all of region \( R \), it implies that for any point \((x, y)\) in \( R \), \( f(x, y) > 0 \). Thus, when graphed, this function creates a surface above the xy-plane.
The significance in integration is evident: the integration collects these positive values across the region. Since each function value is positive, the resulting integral reflects this positivity.

Region R

A region \( R \) in a double integral scenario is simply the area in the xy-plane over which the function is being evaluated. This can be a rectangle, a circle, or any other shape, as long as it is well-defined and bounded.

• When using a double integral, you're essentially looking at how the function \( f(x, y) \) behaves within this region \( R \).
• The choice of region can impact the resulting integral, as it dictates where the function's values will be accumulated.

Hence, understanding the boundaries and area of \( R \) is crucial when calculating the total value of the double integral. The shape and size of \( R \) influence the total accumulation of \( f(x, y) \) over that region.

Accumulation of Values

In integration, especially with functions over a region, accumulation refers to collecting and summing some quantity over a specified area. The double integral \( \iint_{R} f(x, y) \, dA \) signifies the aggregation of the function’s values across region \( R \).

• Each tiny piece of area within \( R \) contributes a value equivalent to the height \( f(x, y) \).
• By summing these minute contributions, you achieve a total that encompasses the entire specified area \( R \).

Therefore, the concept of accumulation in a double integral is central to understanding how the function’s output behaves over \( R \). When the function is positive everywhere, this accumulation results in a positive total sum, as there are no negative values to subtract or cancel out.

Infinitesimal Area dA

The term \( dA \) represents a very small area element in the region \( R \). Imagine breaking down \( R \) into countless tiny squares or rectangles—each one is an infinitesimal area \( dA \).
This minute area is used to help integrate the value of the function over the region. When considering each little section, the function \( f(x, y) \) gives a contribution to the double integral equal to \( f(x, y) \, dA \).

• These infinitesimal areas are the building blocks of the integration process.
• As you add up all these tiny contributions, you find the total integral over the entire region \( R \).

The idea of infinitesimal \( dA \) allows us to meticulously and accurately capture the total effect of \( f(x, y) \) over \( R \), resulting in a sum (integral) that mirrors the function's nature within \( R \).