Problem 57 - Thomas Calculus

Question

In Exercises \(55-60,\) (a) find the spherical coordinate limits for the integral that calculates the volume of the given solid and then (b) evaluate the integral. The solid enclosed by the cardioid of revolution \(\rho=1-\cos \phi\)

Step-by-Step Solution

Answer

The volume of the solid is calculated by using the given limits in the integral.

Step 1: Understand the Problem

We need to find the spherical coordinate limits for the solid defined by the cardioid of revolution \(\rho=1-\cos \phi\), and then use these limits to evaluate an integral for the volume enclosed by this solid.

Step 2: Identifying Spherical Coordinates

In spherical coordinates, the position is defined as \((\rho, \phi, \theta)\), where \(\rho\) is the radial distance, \(\phi\) is the polar angle, and \(\theta\) is the azimuthal angle. The cardioid given is \(\rho=1-\cos \phi\).

Key Concepts

Polar Angle

The polar angle, denoted as \( \phi \) in spherical coordinates, is a crucial component in defining a point in three-dimensional space. It is also referred to as the "zenith angle" and is measured in relation to a fixed axis, usually the positive \( z \)-axis. The range of the polar angle \( \phi \) generally spans from \( 0 \) to \( \pi \). Whether you're analyzing points on a globe or plotting points in a 3D model, understanding the polar angle helps in visualizing and calculating the precise location of the point. For the cardioid of revolution \( \rho=1-\cos \phi \), \( \phi \) acts as a parameter that helps define the shape of the surface that needs to be considered when setting the bounds for the volume integral.

Radial Distance

Radial distance, represented as \( \rho \), indicates how far a point is from the origin in a spherical coordinate system. Imagine a sphere with the origin at its center, and \( \rho \) is the radius connecting this center to any point on the sphere's surface. For the cardioid of revolution \( \rho=1-\cos \phi \), \( \rho \) isn't fixed, but rather varies as \( \phi \) changes. By expressing \( \rho \) in terms of \( \phi \), we can explore how radius changes define the given surface. In this specific scenario, \( \rho=1-\cos \phi \) varies between \( 0 \) and \( 2 \) as \( \phi \) shifts from \( 0 \) to \( \pi \). This variability marks the boundary of the solid, proving integral in volume calculation.

Azimuthal Angle

The azimuthal angle, \( \theta \), is another key spherical coordinate measured in the plane perpendicular to the polar axis. It spans around the \( z \)-axis and ranges from \( 0 \) to \( 2\pi \), resembling longitude lines on the Earth. It contributes to defining a complete spatial orientation by indicating rotation around the vertical axis. While for some solids, \( \theta \) might remain constant, in the case of the cardioid of revolution \( \rho=1-\cos \phi \), \( \theta \) is allowed to vary fully between \( 0 \) and \( 2\pi \), forming a full rotational symmetry. This comprehensive range facilitates the calculation of the entire structure's volume as the solid revolves completely around the \( z \)-axis.

Volume Integral

Calculating a volume integral in spherical coordinates involves integration in terms of \( \rho \), \( \phi \), and \( \theta \). These integrations are useful in determining the size of a solid in three-dimensional space. Using the cardioid \( \rho=1-\cos \phi \), the volume is computed by integrating the function \( \rho^2 \sin \phi \) over the defined intervals for \( \rho \), \( \phi \), and \( \theta \). The limits for \( \rho \) come from the cardioid equation itself, \( \rho=1-\cos \phi \), while \( \phi \) and \( \theta \) have their typical bounds of \([0, \pi]\) and \([0, 2\pi]\), respectively. This triple integral then provides the volume enclosed by the cardioid, revealing how the geometric properties of \( \rho \), \( \phi \), and \( \theta \) interplay to define space.

Cardioid of Revolution

A cardioid of revolution describes a surface created by revolving a cardioid shape around an axis, forming a three-dimensional solid. The equation \( \rho=1-\cos \phi \) outlines this specific cardioid. As \( \phi \) varies, distinct points are established, shaping a revolved surface around the \( z \)-axis. This revolution in spherical coordinates offers a unique, heart-like shape that holds fascinating properties for mathematical and physical analysis. By analytically exploring the cardioid of revolution, students can better grasp how radial variations and revolutions create complex, yet predictable, solid forms. This understanding is essential in many fields, including geometry, physics, and engineering, where such structures often emerge.