91Ó°ÊÓ

Cylindrical coordinates are a way of expressing points in a space using a combination of a radial distance, an angular coordinate, and a vertical position. This system is particularly useful when dealing with problems involving objects with circular symmetry, like cylinders. In cylindrical coordinates, a point in three-dimensional space is defined as

• \(r\): the radial distance from the z-axis.
• \(\theta\): the angle measured counterclockwise from the positive x-axis.
• \(z\): the same vertical distance as in Cartesian coordinates.

These coordinates are related to Cartesian coordinates \((x, y, z)\) by:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(z = z\)

Cylindrical coordinates simplify many integrals because they align better with the symmetry of circular objects. For example, finding volumes of regions related to cylinders or objects with rotational symmetry tends to be more straightforward in these coordinates since \(r\) and \(\theta\) naturally describe circular paths around the z-axis.

The calculation of volume using integrals involves determining how much space an object occupies. In our problem, we're using triple integration, which extends the concept of a double integral to three dimensions. For volume calculations in cylindrical coordinates, the volume element is expressed as \(dV = r \, dr \, d\theta \, dz\). Here:

• \(r \, dr\) represents the radial portion of a doughnut-like ring or circular disk.
• \(d\theta\) describes how that ring extends around the z-axis, covering an entire circle.
• \(dz\) accounts for how the space extends vertically from the base to the top of the figure.

In the given exercise, the limits were set up to only account for the area inside the sphere and outside the cylinder:

• The radial distance \(r\) extends from 1 to \(\sqrt{2 - z^2}\), enclosing the region outside the cylinder.
• \(\theta\) covers 360 degrees, or \(0\) to \(2\pi\), full circle around the z-axis.
• \(z\) stretches from \(-\sqrt{2}\) to \(\sqrt{2}\), encompassing the full vertical extension of the sphere.

So, solving the integral with respect to \(r, \theta,\) and \(z\) systematically carves out the volume accounting for each dimension.

Understanding the intersection between a sphere and a cylinder is central to solving the problem. A sphere is defined by its radius and center point, given in the equation form \(x^2 + y^2 + z^2 = R^2\). A cylinder stretches infinitely along one axis, here the z-axis, and is defined by the equation \(x^2 + y^2 = a^2\) where \(a\) is the cylinder's radius.
In our problem, the sphere's equation \(x^2 + y^2 + z^2 = 2\) means a radius of \(\sqrt{2}\). The cylinder is defined as \(x^2 + y^2 = 1\), having a radius of 1.
When these two surfaces intersect, any volume calculations must address how they overlap.

• The sphere, with a larger radius, completely encloses the cylinder's cross-section at any given point along the z-axis.
• The volume of interest lies between the cylinder's outer surface (\(r = 1\)) and the sphere's outer boundary (\(r = \sqrt{2 - z^2}\)).

By carefully defining our limits in the triple integral setup, we accurately compute only the volume that fits this description, ensuring no part of the interior or exterior unintended to be included in the calculation. This nuanced understanding of overlapping geometries is essential when using integrals to determine complex volumes.