Question

Explain why the following statement is false: "Fubini's Theorem states that \(\int_{a}^{b} \int_{g_{1}(x)}^{g_{2}(x)} f(x, y) d y d x\) \(\int_{a}^{b} \int_{g_{1}(y)}^{g_{2}(y)} f(x, y) d x d y . "\)

Short answer

The statement is false because Fubini's Theorem requires constant integration limits, not dependent on x or y.

Step-by-step solution

Understanding Fubini's Theorem

Fubini's Theorem provides a method to evaluate double integrals in terms of iterated integrals. It specifically applies when a function is integrable on the Cartesian product of two measurable spaces (such as \(\mathbb{R}^2\)). The order of integration (e.g., \(dx\, dy\) or \(dy\, dx\)) is interchangeable under certain conditions when the limits are constants, not functions.

Analyzing the Given Statement

The given statement implies that Fubini's Theorem allows for changing the order of integration from \([g_1(x), g_2(x)]\) with respect to y and then x, to \([g_1(y), g_2(y)]\) with respect to x and then y. However, these are not constant limits; they depend on x and y, respectively.

Identifying the Error

Fubini's Theorem applies only to regions where the limits of integration are constants and the function is integrable within those bounds. When the limits are functions of the variables (non-rectangular region), the order of integration can only be swapped using the formula for changing variables, and even then, additional considerations are needed to ensure the integral bounds correctly describe the region.

Concluding Why the Statement is False

Because Fubini's Theorem requires a specific type of domain (typically a product of intervals with constant bounds), the statement about changing the order of integration in cases where the limits are functions (like \(g_1(x), g_2(x)\) or \(g_1(y), g_2(y)\)) without proper justification or restructuring of the domain is incorrect.

Key concepts

Double Integrals

Double integrals are an extension of single-variable integrals. They are used to compute the volume under a surface over a two-dimensional region. The notation for a double integral is \[ \int \int_{D} f(x, y) \, dA \],where \(D\) is the region of integration in the \(xy\)-plane. Double integrals can be difficult to compute by hand, but they are immensely useful in fields like physics and engineering.
The computation of double integrals involves breaking the region \(D\) into small rectangles, then computing the sum of the volume over these rectangles.

• When \(D\) is a rectangle, it's straightforward to set up the double integral with constant limits.
• However, \(D\) can be a more complex shape, requiring functions to describe the boundaries.

The double integral calculates accumulated quantities across a plane, which is especially helpful when evaluating physical quantities like mass or charge distribution.

Iterated Integrals

Iterated integrals refer to the process of computing a double integral by solving two single integrals in sequence. You solve one integral first, then use its result to solve the second.
This method is made possible by Fubini's Theorem, which allows iterating because it states the double integral over a rectangular region can be computed as two separate integrals.
Consider the general iterated integral:\[\int_{a}^{b} \left( \int_{c}^{d} f(x, y) \, dy \right) dx\]The inner integral is evaluated first, while the outer integral is computed after that result.

• This process effectively layers one integral over the result of the first, like calculating area, then stretching it along another dimension.
• If the region isn’t rectangular, care must be taken to interpret the limits correctly, adjusting for the shape of the domain.

Iterated integrals streamline the integration process, making complex double integrals manageable in many mathematical and practical contexts.

Order of Integration

The order of integration in double integrals is the sequence in which you perform the integration. Typically, this can be either \(dx\, dy\) or \(dy\, dx\).
The order chosen can simplify the problem or make it more complex, depending on the setup of the integral and the functions involved.

• For rectangular regions, the order doesn't matter, provided Fubini's Theorem applies.
• When original limits depend on variables, the process becomes intricate as it may require changing integration order by properly adjusting bounds with respect to the given functions.

When dealing with variable limits, the dependency of one variable over the other dictates how you approach the problem. You may need to convert the double integral setup entirely to make it solvable. Understanding the order of integration helps in visualizing and accurately defining the area under evaluation.

Non-Rectangular Regions

Non-rectangular regions in double integrals are areas where the usual straightforward limits (like constants) are not applicable. Instead, the region boundaries are defined by functions. This adds complexity to evaluating the integral.
When a region isn't rectangular, you must carefully define the upper and lower limits of integration, considering the curves or lines that describe the boundaries.

• These functions determine how the region stretches or compresses over the plane, leading to different integration limits and strategies.
• Think of integrating under a tilted or curved surface, requiring precise setting of both horizontal and vertical bounds.

Evaluating double integrals over non-rectangular regions often requires a step-by-step approach to adjust and correctly interpret these bounds, ensuring an accurate outcome. Proper understanding and manipulation of these variable limits ensure that the integration truly represents the intended area or volume.