Problem 61 - Thomas Calculus

Question

Cylinder and sphere Find the volume of the region cut from the solid cylinder \(x^{2}+y^{2} \leq 1\) by the sphere \(x^{2}+y^{2}+z^{2}=4\)

Step-by-Step Solution

Answer

The volume of the region is \(2\pi\sqrt{3}\).

Step 1: Understand the Problem

The problem involves finding the volume of the intersection between a cylinder and a sphere. The cylinder is given by the inequality \(x^2 + y^2 \leq 1\) and extends infinitely in the \(z\)-direction, while the sphere's equation is \(x^2 + y^2 + z^2 = 4\). We need to visualize where the cylinder lies within the sphere and determine the region of intersection.

Step 2: Set Up Equations of Surfaces

The cylinder equation \(x^2 + y^2 = 1\) represents a circular cross-section with radius 1. The sphere equation \(x^2 + y^2 + z^2 = 4\) represents a sphere centered at the origin with radius 2. The task is to find the overlapping region of these two solids, which will form a cylinder capped by the sphere's surface.

Step 3: Find the Boundaries

To find how far the cylinder extends into the sphere, we set \(x^2 + y^2 = 1\) (the lateral boundary of the cylinder) into the sphere's equation:\[ 1 + z^2 = 4 \]This simplifies to \(z^2 = 3\), giving \(z = \pm\sqrt{3}\). Therefore, the cylinder is cut between \(z = -\sqrt{3}\) and \(z = \sqrt{3}\) by the sphere.

Step 4: Calculate the Volume of the Cylinder Section

The volume of a cylinder of height \(h\) and radius \(r\) is \(\pi r^2 h\). In this problem, \(r = 1\) and \(h = 2\sqrt{3}\) because it ranges from \(-\sqrt{3}\) to \(\sqrt{3}\). Thus, the volume is:\[ V = \pi \times 1^2 \times 2\sqrt{3} = 2\pi\sqrt{3} \]

Step 5: Evaluate the Result

The domain of integration encompassed the whole cylinder, but its upper and lower ends were capped by the spherical boundary at \(z = \pm\sqrt{3}\). Confirm that this volume only considers material inside both the cylinder and the sphere.

Key Concepts

Understanding Cylinders

A cylinder is a three-dimensional shape with two parallel bases connected by a curved surface at a fixed distance from each base. It is crucial to visualize its geometric properties to understand its intersections with other shapes. In mathematics, when you see a cylinder described as having an equation like \(x^2 + y^2 \leq 1\), this describes a circular cross-section in the \(xy\)-plane and suggests that the cylinder extends infinitely along the \(z\)-axis. Therefore, the 'height' of this cylinder isn't limited until specified by additional constraints, such as the presence of another solid body like a sphere.

Exploring the Nature of Spheres

A sphere is a perfectly symmetrical three-dimensional object where every point on its surface is equidistant from its center. The radius is this distance, and it is pivotal in describing the sphere's equation. Consider the equation \(x^2 + y^2 + z^2 = 4\). This represents a sphere centered at the origin (0,0,0) with a radius of 2. Visualizing it helps to understand any intersections with other geometric shapes, as the entire spatial distribution needs to be accounted for when finding volumes, especially in complex intersections with cylinders.

The Process of Volume Integration

Volume integration is a crucial technique to determine the volume of a solid, especially when it involves finding the regions where solids intersect. It's about summing infinitesimally small slices to form an entire volume - imagine stacking numerous thin layers. In this context, you integrate along the \(z\)-axis within the bounds set by intersecting shapes. For example, if we have determined the bounds to be from \(z = -\sqrt{3}\) to \(z = \sqrt{3}\), as the cylinder 'caps' inside the sphere, the volume of this section of the cylinder is computed through integration across these specific limits.

Understanding Geometric Intersection

When two geometric shapes intersect, they form a new region that combines parts of both objects. Understanding the intersection between a cylinder and a sphere involves acknowledging both shapes' intrinsic properties. In our exercise, the cylinder's infinite height is practically limited by the spherical boundary. The intersection itself results in a shape formed by the cylinder culminating into curved caps at both ends, defined by the sphere. Recognizing intersection boundaries is critical because they delineate the actual region of interest when performing calculations such as volume integration.