Problem 11 - Thomas Calculus

Question

Let \(D\) be the region bounded below by the plane \(z=0,\) above by the sphere \(x^{2}+y^{2}+z^{2}=4,\) and on the sides by the cylinder \(x^{2}+y^{2}=1 .\) Set up the triple integrals in cylindrical coordinates that give the volume of \(D\) using the following orders of integration. a. \(d z d r d \theta\) b. \(d r d z d \theta\) c. \(d \theta d z d r\)

Step-by-Step Solution

Answer

Set up triple integrals as: (a) \( \int_{0}^{2\pi} \int_{0}^{1} \int_{0}^{\sqrt{4-r^2}} r\ dz\ dr\ d\theta \), (b) \( \int_{0}^{2\pi} \int_{0}^{\sqrt{4-z^2}} \int_{0}^{1} r\ dr\ dz\ d\theta \), (c) \( \int_{0}^{1} \int_{0}^{\sqrt{4-r^2}} \int_{0}^{2\pi} r\ d\theta\ dz\ dr \).

Step 1: Understanding the Problem

We need to find the volume of the solid region, \(D\), bounded by three surfaces: the plane \(z = 0\), the sphere given by \(x^2 + y^2 + z^2 = 4\), and the cylinder \(x^2 + y^2 = 1\). We will express the triple integrals in cylindrical coordinates: \(x = r\cos\theta, y = r\sin\theta, z = z\). This requires us to set limits for \(z\), \(r\), and \(\theta\) in an appropriate order.

Step 2: Converting Surfaces to Cylindrical Coordinates

Convert the surfaces into cylindrical coordinates: - The cylinder \(x^2 + y^2 = 1\) becomes \(r = 1\).- The sphere \(x^2 + y^2 + z^2 = 4\) becomes \(r^2 + z^2 = 4\).- The plane \(z = 0\) remains the same.These conversions will help set the integration limits.

Step 3: Setting Limits for Integration (dz dr dθ)

For integration order \(dz\ dr\ d\theta\):- \(z\) ranges between its lower bound on the plane \(z = 0\) and the upper bound on the sphere: \(z = \sqrt{4 - r^2}\).- \(r\) ranges from 0 to the boundary of the cylinder: \(r = 1\).- \(\theta\) ranges from 0 to \(2\pi\).The integral is \[ \int_{0}^{2\pi} \int_{0}^{1} \int_{0}^{\sqrt{4-r^2}} r\ dz\ dr\ d\theta \].

Step 4: Setting Limits for Integration (dr dz dθ)

For integration order \(dr\ dz\ d\theta\):- \(r\) ranges from 0 to the cylinder \(r = 1\).- \(z\) for a fixed \(r\) ranges from the plane \(z = 0\) to the sphere \(z = \sqrt{4 - r^2}\).- \(\theta\) ranges from 0 to \(2\pi\).The integral is \[ \int_{0}^{2\pi} \int_{0}^{\sqrt{4-z^2}} \int_{0}^{1} r\ dr\ dz\ d\theta \].

Step 5: Setting Limits for Integration (dθ dz dr)

For integration order \(d\theta\ dz\ dr\):- \(\theta\) ranges from 0 to \(2\pi\).- \(z\) for a fixed \(\theta\) ranges from \(z = 0\) to \(z = \sqrt{4 - r^2}\).- \(r\) for the given \(z\) ranges from 0 to 1.The integral is \[ \int_{0}^{1} \int_{0}^{\sqrt{4-r^2}} \int_{0}^{2\pi} r\ d\theta\ dz\ dr \].

Key Concepts

Triple Integrals

When dealing with volume calculations in three-dimensional space, triple integrals are an invaluable tool. Triple integrals allow us to integrate over a three-dimensional region, which is necessary for finding volume, among other things. In cylindrical coordinates, the triple integral is expressed in terms of the variables \( r \), \( \theta \), and \( z \), which correspond to the radius, angle, and height, respectively.

For our region \( D \), which is bounded by the plane \( z = 0 \), a sphere, and a cylinder, setting up the triple integral involves choosing an order of integration that suits the geometric properties of the region. The different orders of integration - \( dz\ dr\ d\theta \), \( dr\ dz\ d\theta \), and \( d\theta\ dz\ dr \) - offer flexibility in evaluating the integral based on practical or simplifying considerations.

The general form of a triple integral in cylindrical coordinates becomes:

• \( \int \int \int \ f(r, \theta, z)\ r\ dz\ dr\ d\theta \), where the limits of integration for \( z \), \( r \), and \( \theta \) depend on the boundaries of the region.

Understanding how to switch and set these limits is critical to the execution of such integrals for calculating volumes.

Volume Calculation

The task of calculating a volume of a 3D region bounded by multiple surfaces using triple integrals can seem daunting at first, but breaking it down makes it more manageable. After recognizing boundaries and converting them to cylindrical coordinates, the next step is to determine the integration limits.

The given problem requires us to consider the region \( D \) under certain surfaces. The volume of \( D \) is explored through integrating over the solid from the bottom plane up to the bounding sphere, while being restricted by the cylindrical sides. It's crucial to:

• Identify the lower and upper bounds for each dimension: \( z \), \( r \), and \( \theta \).
• Select an integration order that aligns with the geometric constraints of the problem.

Using the cylindrical coordinates:

• The radius \( r \) is bounded by the cylinder \( x^2 + y^2 = 1 \) and takes values from 0 to 1.
• The height \( z \) varies from the plane, \( z = 0 \), to the curved surface of the sphere: \( z = \sqrt{4-r^2} \).
• The angle \( \theta \) naturally spans from 0 to \( 2\pi \).

Each of these boundaries plays a role in directly influencing the computation of the volume.

Spherical Coordinates

While primarily solving the problem using cylindrical coordinates, it's beneficial to also understand spherical coordinates due to their common use in solving volume integrals involving spheres. Spherical coordinates are an alternative that can simplify problems involving symmetrical shapes like spheres.

Spherical coordinates consist of:

• \( \rho \): the radius or distance from the origin.
• \( \phi \): the angle down from the positive z-axis.
• \( \theta \): the same angular component used in cylindrical coordinates.

In this scenario, the critical aspect to note is that the sphere has an equation \( x^2 + y^2 + z^2 = 4 \), which aligns naturally with the spherical form \( \rho^2 = 4 \), or \( \rho = 2 \). When dealing with the spherical region in Cartesian coordinates, using spherical coordinates could potentially simplify integration depending on the problem structure.

However, for this problem specifically focusing on cylindrical coordinates provides the clearest path to an effective solution, considering the cylindrical boundary directly embedded in the problem statement.