Problem 43 - Thomas Calculus

Question

Centroid Find the centroid of the portion of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) that lies in the first octant.

Step-by-Step Solution

Answer

Centroid: \( \left( \frac{3a}{8}, \frac{3a}{8}, \frac{3a}{8} \right) \).

Step 1: Understand the Sphere

The given equation of the sphere is \(x^{2} + y^{2} + z^{2} = a^{2}\). This represents a sphere centered at the origin with a radius of \(a\).

Step 2: Define the Portion in the First Octant

The first octant is defined by the region where \(x \, \geq \, 0\), \(y \, \geq \, 0\), and \(z \, \geq \, 0\). We only consider the part of the sphere that satisfies these conditions.

Step 3: Identify Coordinate Ranges

For integration, the ranges in the first octant are: \(0 \leq x \leq a\), \(0 \leq y \leq \, \sqrt{a^2 - x^2}\), and \(0 \leq z \leq \, \sqrt{a^2 - x^2 - y^2}\).

Step 4: Integrate to Find Volume

The volume \(V\) of the first octant portion can be found using triple integration: \[V = \int_{0}^{a} \int_{0}^{\sqrt{a^2 - x^2}} \int_{0}^{\sqrt{a^2 - x^2 - y^2}} \, dz \, dy \, dx.\]Solving this gives \(V = \frac{1}{8} \cdot \frac{4}{3} \pi a^3 = \frac{\pi a^3}{6}\).

Step 5: Compute Coordinates of Centroid

The coordinates of the centroid \((\bar{x}, \bar{y}, \bar{z})\) are given by:\[\bar{x} = \frac{1}{V} \int_{0}^{a} \int_{0}^{\sqrt{a^2 - x^2}} \int_{0}^{\sqrt{a^2 - x^2 - y^2}} x \, dz \, dy \, dx,\]\[\bar{y} = \frac{1}{V} \int_{0}^{a} \int_{0}^{\sqrt{a^2 - x^2}} \int_{0}^{\sqrt{a^2 - x^2 - y^2}} y \, dz \, dy \, dx,\]\[\bar{z} = \frac{1}{V} \int_{0}^{a} \int_{0}^{\sqrt{a^2 - x^2}} \int_{0}^{\sqrt{a^2 - x^2 - y^2}} z \, dz \, dy \, dx.\]

Step 6: Calculate Each Integral

After performing the integration, you find:\[\bar{x} = \bar{y} = \bar{z} = \frac{3a}{8}.\] This is because the symmetry in the first octant makes the computation yield the same result for each coordinate.

Step 7: Verify Final Result

Given the symmetry of the sphere and the constraints, the centroid \( \left( \frac{3a}{8}, \frac{3a}{8}, \frac{3a}{8} \right) \) is correct for the first octant, reaffirming symmetry is maintained.

Key Concepts

Triple Integration

Triple integration is a mathematical technique used to calculate volumes and other properties of three-dimensional shapes. In this exercise, triple integration is employed to find the volume of the portion of the sphere that lies within the first octant. Understanding this requires recognizing how each integral
works through the layers of a 3D shape from outer to inner limits:

• The outermost integral deals with the "x" direction.
• The middle integral handles the "y" direction, dependent on the "x" value.
• The innermost integral involves the "z" direction, influenced by both "x" and "y" values.

To successfully perform triple integration, each limit of integration must be determined with care based on the shape's criteria. Here,
the bounds in Cartesian coordinates are based on the sphere's equation and its restriction to the first octant. Integrating within these limits allows for calculation of the specific region's total volume, a necessary step for locating the centroid.

First Octant

The first octant is a fundamental concept in three-dimensional geometry. When referring to the first octant, it is the section of the coordinate system where all three variables—x, y, and z—are positive. It essentially divides the 3D space into eight symmetrical parts.
For a sphere centered at the origin, the first octant is filled with a specific portion of the sphere. Within this region, the equations that limit each axis's integration reflect the characteristics
of the given surface:

• The x-axis ranges from 0 to the radius of the sphere, "a".
• The y-axis is dependent on "x" and goes from 0 to \( \sqrt{a^2 - x^2} \).
• The z-axis builds upon the y-range, extending from 0 to \( \sqrt{a^2 - x^2 - y^2} \).

Focusing on the first octant simplifies integration, thereby offering clearer calculations. This focus allows students to apply symmetry to make easier computations and arrive at conclusions such as centroid determinations.

Symmetry

In mathematical terms, symmetry refers to a certain invariant property where an object looks the same from various perspectives or orientations. In this exercise, symmetry plays a significant role in simplifying the calculations for finding the centroid.
The sphere inherently possesses uniform symmetry in all directions, and this property is preserved even when it is constrained to the first octant. Because of symmetry, both centroid coordinates, \((\bar{x}, \bar{y}, \bar{z})\), end up being equal; particularly,
they are constrained by just one calculation:

• \( \bar{x} = \bar{y} = \bar{z} \)

This results from the balancing of integral relations, where all the axes have similar limits once the symmetric portion is taken into account. Symmetry is thus not only an attribute of the geometry but a practical tool in finding results effortlessly.

Volume of a Solid

The volume of a solid is a crucial characteristic that describes the amount of space an object occupies in three dimensions. In the context of the sphere residing in the first octant, volume calculations must consider the sphere's limitations.
By integrating over the defined bounds,

• Triple integrals assist in encapsulating the total space enclosed within the solid's constraints.
• The derived volume formula for the part of the sphere in the first octant, \( V = \frac{\pi a^3}{6} \), is computed by evaluating the specified range of integration.

Understanding the volume of this segment helps identify where the centroid lies, offering insight into the shape's equilibrium point. In turn, the volume serves as a foundational element in other calculations regarding a solid's calculus-based properties, such as inertia, center of mass, and density.