Problem 34 - Calculus Early Transcendentals

Question

The centroid of a surface \(\sigma\) is defined by $$ \quad \bar{x}=\frac{\iint_{\sigma} x d S}{\operatorname{arcaof} \sigma}, \quad \bar{y}=\frac{\iint_{\sigma} y d S}{\operatorname{arcaof} \sigma}, \quad \bar{z}=\frac{\iint_{\sigma} z d S}{\operatorname{arcaof} \sigma} $$ Find the centroid of the surface. The portion of the sphere \(x^{2}+y^{2}+z^{2}=4\) above the plane \(z=1\)

Step-by-Step Solution

Answer

The centroid of the surface is at (0, 0, 0.75).

Step 1: Identify the surface

We are given a sphere with equation \(x^2 + y^2 + z^2 = 4\) and need to find the portion of this sphere above the plane \(z = 1\). This corresponds to a spherical cap.

Step 2: Express the area element

In spherical coordinates, the surface element on a sphere is \(dS = R^2 \sin\theta \, d\theta \, d\phi\), where \(R = 2\) is the radius of the sphere, \(0 \leq \phi < 2\pi\), and \(\theta\) varies from \(0\) to the angle at which \(z = 1\).

Step 3: Find angle limit \(\theta\) for \(z = 1\)

The equation \(z = R \cos\theta = 1\) gives \(\cos\theta = \frac{1}{2}\). Therefore, \(\theta = \frac{\pi}{3}\) is the upper limit for the cap's portion.

Step 4: Calculate the area of the surface \(\sigma\)

Integrate the expression \(R^2 \sin\theta\) over the specified limits: \[\operatorname{area}(\sigma) = \int_{0}^{2\pi}\int_{0}^{\pi/3} 4 \sin\theta \, d\theta \, d\phi = 4 \left(2\pi\right) \left(1 - \cos\left(\frac{\pi}{3}\right)\right) = 4 \left(2\pi\right) \left(1 - \frac{1}{2}\right) = 4\pi\].

Step 5: Calculate \(\bar{x}\) and \(\bar{y}\)

Since the surface is symmetric about the z-axis, the integrals for \(\bar{x}\) and \(\bar{y}\) over the spherical cap will be zero. Thus, \(\bar{x} = 0\) and \(\bar{y} = 0\).

Step 6: Calculate \(\bar{z}\)

Calculate \(\bar{z}\) using the formula:\[\bar{z} = \frac{\iint_{\sigma} z \, dS}{\operatorname{area}(\sigma)} = \frac{\int_{0}^{2\pi}\int_{0}^{\pi/3} 2 \cos\theta \cdot 4 \sin\theta \, d\theta \, d\phi}{4\pi}\].The innermost integral is:\[8 \int_{0}^{\pi/3} \cos\theta \sin\theta \, d\theta = 8 \left[\frac{1}{2} \sin^2\theta\right]_{0}^{\pi/3} = 8 \cdot \frac{1}{2} \left(\left(\frac{\sqrt{3}}{2}\right)^2 - 0\right) = 2 \cdot 3 = 3\].Therefore, \(\bar{z} = \frac{3 \cdot 2 \pi}{4\pi} = \frac{3}{4}\).

Key Concepts

Spherical Coordinates

Spherical coordinates offer a powerful way to express points in three-dimensional space. Unlike Cartesian coordinates, which use \(x, y, z\) positions, spherical coordinates describe a point using:

• Radius \(R\): Distance from the origin to the point
• Polar angle \(\theta\): Angle measured from the positive z-axis
• Azimuthal angle \(\phi\): Angle measured in the xy-plane from the positive x-axis

For the sphere considered in the exercise, \(R = 2\), which is the sphere's radius. The limits for \(\phi\) and \(\theta\) let us cover the spherical cap, the region above the plane \(z = 1\). This unique way of representing areas makes calculations, such as those for surface integrals, more manageable in problems involving spheres.

Surface Integral

A surface integral extends the concept of an integral, summing values over a surface in space. In spherical coordinates, the surface element \(dS\) is expressed as \(R^2 \sin\theta \, d\theta \, d\phi\). This accounts for variations in both the polar and azimuthal angles.

• The integration over \(\phi\) typically ranges from \(0\) to \(2\pi\) to handle full rotations around the z-axis.
• For \(\theta\), limits depend on the problem setup, often defined by intersections with other shapes, like a plane.

In our example, calculating the surface integral over the spherical cap requires integrating \(z \, dS\), where the integrand \(z = 2\cos\theta\) because of the spherical constraint. This process allows us to find relevant geometric properties, such as the centroid of the surface.

Spherical Cap

A spherical cap is a portion of a sphere cut by a plane. In the problem, the spherical cap consists of the part of the sphere \(x^2 + y^2 + z^2 = 4\) above the plane \(z = 1\).

• The defining characteristic is the angle \(\theta\), calculated from \(\cos\theta = \frac{1}{2}\), giving \(\theta = \frac{\pi}{3}\).
• The radius of the base of the cap isn't vital for the centroid calculation but knowing its range enables precise integration.

Finding the area of the spherical cap requires integrating the surface element \(dS\) over \(\phi\) from \(0\) to \(2\pi\) and \(\theta\) from \(0\) to \(\frac{\pi}{3}\). This reveals the area essential for centroid calculations.

Symmetry

Symmetry plays a crucial role in simplifying calculations. For our spherical cap, symmetry can reduce computational effort related to centroids.

• In this problem, symmetry about the z-axis ensures that the x and y components of the centroid equal zero: \(\bar{x} = 0\) and \(\bar{y} = 0\).
• Only \(\bar{z}\) requires calculation since symmetrical properties indicate the centroid will lie along the z-axis.

Natural symmetries like these mean that instead of separately calculating \(\bar{x}\), \(\bar{y}\), and \(\bar{z}\), we can focus on just one component, hence simplifying the process and saving time. This emphasizes the power of symmetry in mathematical problem-solving.