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The example in your image begins by describing a set \( A \) in an \( n \)-dimensional space and defines an integral over the set \( A \) using the notation:

\[
Q(A) = \int_A \dots \int dx_1 \dots dx_n
\]

This represents a multiple integral over the set \( A \) in \( n \)-dimensional space, where the integrals are taken with respect to the variables \( x_1, \dots, x_n \).

The example goes on to specify the set \( A \) as:

\[
A = \{ (x_1, x_2, \dots, x_n) : a_1 \leq x_1 \leq b_1, \dots, a_n \leq x_n \leq b_n \}
\]

This set defines a rectangular region in \( n \)-dimensional space with bounds \( a_i \leq x_i \leq b_i \) for each dimension \( i \).

The final step would likely involve solving the integral for this set \( A \), which typically results in:

\[
Q(A) = \prod_{i=1}^{n} (b_i - a_i)
\]

This result represents the volume (or measure) of the set \( A \) in \( n \)-dimensional space.

Do you want me to go into more details, such as solving an example, or explain more about the concept?

Here are 5 related questions to expand this topic:
1. How do you interpret integrals in \( n \)-dimensional spaces geometrically?
2. What is the difference between volume in 3D space and in higher dimensions?
3. How do bounds affect the result of a multiple integral?
4. Can this concept be applied to non-rectangular regions in \( n \)-dimensional space?
5. How does this integral generalize to different types of measure, like surface integrals?

**Tip**: When working with multiple integrals, ensure that the limits of integration are properly aligned with the region's boundaries to avoid mistakes in calculating the volume or area.

Let A be a set in n-dimensional space and let Q(A) = ∫...∫dx_1...dx_n. If A is the set defined by a_1 ≤ x_1 ≤ b_1, ..., a_n ≤ x_n ≤ b_n, what is the result of Q(A)?

This example explains how to calculate the volume of a set in n-dimensional space using multiple integrals. The set A is defined by bounds for each dimension, and the volume is found through a product of the differences in bounds. Explore concepts in multivariable calculus and n-dimensional integration.

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