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Finding the volume of solids is an essential concept in calculus, especially when dealing with three-dimensional shapes. This becomes particularly interesting when the solids are formed by revolving a two-dimensional region around an axis. These types of problems often arise in calculus and require a deeper understanding of geometric transformations. When we talk about these volumes formed by rotation, we are looking at the solid of revolution. This is a process of taking a region defined by curves and revolving it to create a three-dimensional solid. For instance, the region between the curves can form unique shapes like cylinders or complex structures depending on the boundaries and axis of rotation.

Revolving regions is the method used to create a solid by rotating a region around an axis. In the given problem, the region is revolved around the y-axis. This concept is essential because it allows us to transition from understanding basic areas to the more complex notion of volume.
The region in question is determined by the curves and lines in the xy-plane. Here, the curves are given by the equations \(y=2x\) and \(y=\frac{x}{2}\). The line \(x=1\) completes the boundary of the region. When these boundaries are revolved around the y-axis, they form a cylindrical shape. By revolving the region, we create concentric cylindrical shells, like the layers of an onion, that contribute to the overall volume of the solid.

• Understand the curves or lines forming the boundaries of the region.
• Identify the axis of rotation, which in this case is the y-axis.
• Visualize the three-dimensional shape resulting from this rotation.

Solving calculus problems, particularly those involving volumes of revolution, requires a series of methodical steps. It's crucial to break down the problem into manageable parts and understand each step clearly.
Firstly, identify the region to be revolved by the curves and boundaries given. Determine the limits of integration, typically where the curves intersect or the bounds are explicitly given. Then, choose a method suitable for volume calculation, such as the shell method, which is well-suited for rotation about the y-axis in this scenario.
By setting up the shell method's integral, each component of the solid is considered systematically:

• "Radius" defined by the distance from the rotation axis.
• "Height" as the difference between the functions of the curves.
• Integration over the defined range to accumulate the volume of shells.

Approaching the problem in these organized steps ensures no part of the problem is overlooked, offering a clear pathway from setup to solution.

Integral calculation is the backbone of determining volumes when revolving regions. Through integration, we can efficiently find the accumulated sum of an infinite number of infinitesimally small parts—in this case, the cylindrical shells.
For the given region, the shell method's integral is formulated as \[ V = 2\pi \int_{0}^{1} x \left(\frac{3x}{2}\right) \, dx \]. Solving this integral requires understanding both the setup and the technique of integration itself.
The steps involved include:

• Substitute and simplify expressions into the integral (e.g., height = \(\frac{3x}{2}\) and radius = \(x\)).
• Integrate the expression \(\frac{3}{2} x^2\) from the lower bound of 0 to the upper bound of 1.
• Solve to find the antiderivative and evaluate the definite integral.
• Multiply the result by \(2\pi\) to account for the revolution.

This step-by-step integration provides the precise volume of the solid, illustrating how the abstract notion of integration solidifies in tangible calculations.