Use the shell method to find the volumes of the solids...

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines about the $y$ -axis.
$$y=3 /(2 \sqrt{x}), \quad y=0, \quad x=1, \quad x=4$$

Key Concepts

Shell Method

The shell method is an integration technique used to calculate the volume of a solid of revolution. Instead of slicing the solid into disks or washers, it breaks the region into cylindrical shells. By integrating the lateral surface areas of these shells along the axis of revolution, one can determine the total volume of the solid.

Volume of Revolution

Volumes of revolution are obtained by rotating a two-dimensional region around a given axis. The resulting three-dimensional object has a volume that can be computed using integration methods such as the shell method or the washer/disk method. Understanding this concept is crucial for transitioning from a geometric area to a spatial volume.

Setting Up the Integral

Setting up the integral involves expressing the volume element in terms of a variable that runs over the region. This requires identifying the radius, height, and thickness of the cylindrical shells in the shell method, and determining the correct limits of integration. Proper setup ensures that the integral correctly represents the cumulative contribution of each shell to the overall volume.

Definite Integration

Definite integration is used to evaluate the total volume by summing up the contributions from all the infinitesimally thin shells over a specified interval. This process involves calculating the antiderivative and applying the limits of integration. Mastery of definite integrals is essential for quantifying the volume of solids generated by rotation.

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