Calculate the volume of the portion of cylinder shown in...

Calculate the volume of the portion of cylinder shown in Fig. 1-21.
Set the $x_{1}$-plane on the base of the cylinder and the origin on the axis. Write, the element of volume of height $h$ as $h d x d y$, with
$$
h=\frac{H}{2}\left(1-\frac{x}{R}\right)
$$

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Key Concepts

Coordinate Systems and Change of Variables

Choosing an appropriate coordinate system or applying a change of variables can simplify the integration process, especially when the base region has a regular shape such as a circle. This technique transforms the integral into a more manageable form by aligning with the geometry of the region.

Slicing Method

The slicing method involves dividing the solid into thin slices, each with a small thickness, and approximating the volume by summing the volumes of these slices. When the height of each slice varies with position, the method requires integrating the height function across the region.

Volume Element

The volume element represents an infinitesimally small part of the solid whose volume is computed by integration. It is expressed as the product of an area element in the base and the height of the solid at that point, allowing the sum (integral) of these elements to yield the total volume.

Double Integration

Double integration is the process of summing values over a two-dimensional region, typically employed to compute volumes when the height of a solid is given as a function over its base. This method involves integrating the height function over the area of the base to accumulate the total volume.

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