Problem 41 - Thomas Calculus

Question

Let \(D\) be the smaller cap cut from a solid ball of radius 2 units by a plane 1 unit from the center of the sphere. Express the volume of \(D\) as an iterated triple integral in (a) spherical, (b) cylindrical, and (c) rectangular coordinates. Then (d) find the volume by evaluating one of the three triple integrals.

Step-by-Step Solution

Answer

Evaluate the spherical integral: the easiest method for finding cap \(D\)'s volume.

Step 1: Understanding the Geometry: Sphere and Cap

Consider a sphere centered at the origin with radius 2 units. A plane cuts through the sphere at 1 unit from the center, parallel to the sphere's equator, creating a spherical cap. The goal is to express and evaluate the volume of this smaller cap, denoted by \(D\).

Step 2: Spherical Coordinates Setup

In spherical coordinates, a point \((x, y, z)\) is represented as \((\rho, \theta, \phi)\), where \(\rho\) is the radial distance, \(\theta\) is the azimuthal angle, and \(\phi\) is the polar angle. The volume element \(dV\) in spherical coordinates is \(\rho^2 \sin\phi \, d\rho \, d\theta \, d\phi\). For the cap, \(\rho\) ranges from 0 to 2, \(\theta\) from 0 to \(2\pi\), and \(\phi\) from 0 to the angle corresponding to \(z = 1\).

Step 3: Determine Spherical Limits

The plane at \(z = 1\) intersects the sphere at \(z = \rho \cos\phi = 1\), giving \(\cos\phi = \frac{1}{\rho}\). Thus, \(\phi\) ranges from 0 to \(\cos^{-1}\left(\frac{1}{2}\right)\). The triple integral in spherical coordinates is \[\int_0^{2\pi} \int_0^{\cos^{-1}\left(\frac{1}{2}\right)} \int_0^2 \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta.\]

Step 4: Cylindrical Coordinates Setup

In cylindrical coordinates, a point is \((r, \theta, z)\), where \(r\) is the radial distance in the \(xy\)-plane, \(\theta\) the azimuthal angle, and \(z\) is the height. The volume element is \(r \, dr \, d\theta \, dz\). For the cap, \(r\) ranges from 0 to \(\sqrt{3}\), \(\theta\) from 0 to \(2\pi\), and \(z\) from \(\sqrt{4 - r^2}\) to 1.

Step 5: Express Cylindrical Triple Integral

The integration limits in cylindrical coordinates reflect the ranges described:\[\int_0^{2\pi} \int_0^{\sqrt{3}} \int_{\sqrt{4 - r^2}}^1 r \, dz \, dr \, d\theta.\]

Step 6: Rectangular Coordinates Setup

In rectangular coordinates, use \((x, y, z)\) with the volume element \(dx \, dy \, dz\). The cap is bound by \(x^2 + y^2 + z^2 \leq 4\) and \(x^2 + y^2 + (z - 1)^2 \geq 1\). Integration is typically the most complex in this coordinate system for spheres due to these inequalities.

Step 7: Rectangular Integration Limits

Express the limits while recognizing the challenges. It's advised to visualize the region or use multiple integrals:\[\int_{-\sqrt{3}}^{\sqrt{3}} \int_{-\sqrt{3 - x^2}}^{\sqrt{3 - x^2}} \int_{\sqrt{4 - x^2 - y^2}}^1 \, dz \, dy \, dx.\]

Step 8: Choice of Integral for Computation

The symmetrical arrangement and simpler integration rules make spherical coordinates the most straightforward for numerical evaluation. Evaluating the spherical triple integral gives the volume of cap \(D\).

Step 9: Evaluation of Spherical Triple Integral

Perform the integration:1. Integrate \(\rho^2\) with respect to \(\rho\) first from 0 to 2.2. Integrate \(\sin\phi\) with respect to \(\phi\) from 0 to \(\cos^{-1}\left(\frac{1}{2}\right)\).3. Integrate the result with respect to \(\theta\) from 0 to \(2\pi\).This evaluates the volume of cap \(D\).

Step 10: Compute Final Volume

After integration, calculate to find numerical volume. Ensuring each integration step correctly applies the limits and fundamental theorems yields the result.

Key Concepts

Spherical Coordinates

Spherical coordinates are a way to represent points in a three-dimensional space using three values: radial distance \(\rho\), azimuthal angle \(\theta\), and polar angle \(\phi\). These values are respectively like how far out from a central point you are, the angle around the axis like longitude, and the angle from the vertical axis, kind of like latitude.

In the context of triple integrals, these coordinates are particularly useful when dealing with spherical objects, like in our cap problem. The formula for the volume element is \(dV = \rho^2 \sin\phi \, d\rho \, d\theta \, d\phi\). In our problem, to find the volume of the spherical cap, the integral is set up as \[\int_0^{2\pi} \int_0^{\cos^{-1}\left(\frac{1}{2}\right)} \int_0^2 \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta,\] which requires understanding each variable's range precisely.

• \(\rho\): the distance from the origin to any point on the sphere, varies from 0 to 2.
• \(\theta\): a complete rotation around the axis, goes from 0 to \(2\pi\).
• \(\phi\): starts from the top and goes down, ranges from 0 to \(\cos^{-1}\left(\frac{1}{2}\right)\) to accommodate the plane intersection at \(z=1\).

Thus, using spherical coordinates helps streamline calculating complex volumes such as spherical caps.

Cylindrical Coordinates

Cylindrical coordinates offer a blend of polar coordinates and a linear coordinate similar to rectangular coordinates, ideal for shapes familiar in the context of cylinders. A point in this system is described by \((r, \theta, z)\), where \(r\) is the radius in the \(xy\)-plane, \(\theta\) is the angular position, and \(z\) is the height.

For our cap problem, cylindrical coordinates are not as natural as spherical ones, but they still provide a useful perspective. The volume element in this system is \(r \, dr \, d\theta \, dz\). For the cap's volume calculation: \[\int_0^{2\pi} \int_0^{\sqrt{3}} \int_{\sqrt{4 - r^2}}^1 r \, dz \, dr \, d\theta,\]this involves setting limits carefully:

• \(r\): measurable from the center to the edge in the plane, capped at \(\sqrt{3}\).
• \(\theta\): the angle around, again from 0 to \(2\pi\).
• \(z\): vertical boundary is from the base of the cap at \(\sqrt{4 - r^2}\) to its top at \(z = 1\).

Such coordinates pivot between cylindrical shapes and spherical angles, providing versatility in various scenarios, though potentially requiring more complex algebra.

Rectangular Coordinates

Rectangular coordinates form the most intuitive system for many, using \((x, y, z)\) to describe any point in space with a straightforward grid-like overlay. Though straightforward, this system becomes challenging for triple integrals dealing with round shapes like spheres, due to non-linear limits.

To handle the triple integral for our spherical cap with rectangular coordinates, it's necessary to navigate inequalities:

• The sphere's boundary is given by \(x^2 + y^2 + z^2 \leq 4\), ensuring points lie within the sphere.
• The additional condition \(x^2 + y^2 + (z - 1)^2 \geq 1\) defines the cut created by the plane.

This drags integration into real complexity, often needing visualization aids or computational software: \[\int_{-\sqrt{3}}^{\sqrt{3}} \int_{-\sqrt{3 - x^2}}^{\sqrt{3 - x^2}} \int_{\sqrt{4 - x^2 - y^2}}^1 \, dz \, dy \, dx.\] Mastering rectangular coordinates for spherical regions demands a deep comprehension of how z-values translate through the sphere's and cap's boundaries. Though they offer ease in many contexts, here they serve more as a mathematic feat requiring greater analytical insight.