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When we examine a solid's cross-sectional area, we're looking at slices of the solid taken at various points along a specific axis—in this case, the x-axis. To determine the cross-sectional area, you need to imagine cutting the solid with parallel planes. Each slice or cut gives us a two-dimensional shape. For every value of x within an interval \[a, b\], the cross-sectional area is denoted by \A(x)\.
Understanding the concept of cross-sectional area is crucial in calculating the volume of irregular solids. By knowing the shape and area of each cross-section, you can imagine stacking up these areas to form the entire volume of the original solid. For a solid with a known cross-sectional shape, like a square or a rectangle, calculating its volume becomes a matter of integrating these areas along the x-axis over the interval from \a\ to \b\.

A rectangular parallelepiped is a geometric term often used to describe a box-shaped structure. It is defined by having three sets of parallel and equal-length edges, which means it has six rectangular faces, all right angles. Think of a shoe box or a cereal box—these are everyday examples of rectangular parallelepipeds.
In the exercise, the crucial factor is that these solids have uniformity in shape across their cross-sections. This characteristic indicates that if cross-sections are squares and remain constant in size (no variation along the x-axis), when these squares are stacked along the axis, they result in a rectangular parallelepiped. Consequently, the absence of any change in side length across the range confirms its box-like structure, making it a straightforward example of this geometric form.

A square cross-section means that when you slice through the solid at any point along the x-axis, you will find a square shape. The side length of this square is denoted by \[s(x)\], which refers to a constant length in this context.
If each cross-section of the solid uniformly maintains the shape and dimensions of a square across every possible cut perpendicular to the x-axis, it suggests that these squares are identical as they extend through the solid. This uniformity in shape and dimensional consistency confirms that stacking such squares results in a solid with all equivalent faces in the section, unequivocally forming a rectangular box or parallelepiped.
This constancy is key because if the side length changed with each cut, the overall solid could take on a more complex or irregular form. But with a stationary side length, the regularity of squares confirms the solid as a box.