Problem 70 - Thomas Calculus

Question

Find the centroid of the solid bounded above by the sphere \(\rho=a\) and below by the cone \(\phi=\pi / 4 .\)

Step-by-Step Solution

Answer

The centroid lies symmetrically along the z-axis.

Step 1: Identify the Solid

The solid is bounded above by a sphere with equation \( \rho = a \), which implies it is a sphere of radius \( a \). The boundary below is defined by the cone \( \phi = \frac{\pi}{4} \), which is an infinite cone with a half-angle of \( \frac{\pi}{4} \). We need the region between these two surfaces.

Step 2: Convert Boundaries to Cylindrical Coordinates

To work with this region, we use spherical coordinates where \( \rho \) is the radial distance, \( \theta \) is the azimuthal angle, and \( \phi \) is the polar angle. The cone's boundary, \( \phi = \frac{\pi}{4} \), and the sphere's, \( \rho = a \).

Key Concepts

spherical coordinates

Spherical coordinates are a system for representing points in three-dimensional space using three parameters. Unlike the Cartesian coordinate system, which uses perpendicular axes (x, y, z), spherical coordinates define a point by:

• Radial Distance (\( \rho \)): The distance from the origin to the point.
• Polar Angle (\( \phi \)): The angle measured from the positive z-axis down to the radius vector of a point.
• Azimuthal Angle (\( \theta \)): The angle in the x-y plane from the positive x-axis.

Calculating objects like spheres or cones becomes easier with this system, due to its ability to easily express radial symmetry. For example, the sphere \( \rho = a \) represents a surface at a constant radial distance, while the cone is represented by a constant angle \( \phi = \frac{\pi}{4} \). By describing surfaces and volumes this way, understanding complex geometries becomes more manageable.

sphere

A sphere in spherical coordinates is particularly simple to define. The sphere centered at the origin with radius \( a \) is represented uniquely by the equation \( \rho = a \). This tells us that all points on the sphere are the same distance \( a \) from the origin. The spherical coordinate system shines because it naturally aligns with the sphere's radial symmetry. Thus, exploring spheres makes spherical coordinates a convenient choice compared to other coordinate systems. Since we only deal with a constant radial distance in the equation, navigating or calculating volumes and surfaces of spheres becomes intuitive and lacks the complexity often found in Cartesian coordinates.

cone

In this problem, the cone is represented by the polar angle equation \( \phi = \frac{\pi}{4} \). A cone in spherical coordinates is defined by a fixed angle, known as the half-angle, between its axis (z-axis) and its surface. This is different from Cartesian representation, where one deals with simultaneous variables.The cone's surface angle of \( \frac{\pi}{4} \) indicates a 45° angle from the vertical z-axis, corresponding to a symmetrical opening that expands outward as the radial distance increases. Such a configuration simplifies the intersection with other shapes culturally, like spheres, aiding problem-solving regarding their boundaries.

calculus

Calculus plays a key role in finding the centroid, especially when dealing with three-dimensional objects like spheres and cones. Here, integrating within spherical coordinates helps calculate volumes and other properties. To find the centroid, we usually need to compute triple integrals. These integrals gather information over the shape's volume to determine average position points, such as the center of mass or centroid. A multidimensional calculus approach is necessary as it utilizes spherical coordinates, offering a uniform way to handle the symmetry effortlessly. Calculus techniques allow us to simplify these complex integrations by breaking down the integral into simpler parts aligning with the geometrical features of the objects involved. With this analytical power, finding centroids becomes precise and structured, crucial in solving real-world physical problems.