Find The Average Height Of A Hemisphere Above The Disk

Imagine slicing a perfectly round dome into countless horizontal layers, each infinitesimally thin. The average height of this dome, a hemisphere, isn't simply half its radius. It requires a more thoughtful approach, blending geometry and calculus to arrive at the correct answer. Let's embark on this journey to discover this average height, exploring the concepts and calculations involved in detail.

Delving into the Hemisphere

A hemisphere, by definition, is half of a sphere. We'll consider a hemisphere of radius R sitting atop a disk of the same radius, centered at the origin of a 3D coordinate system. Our goal is to determine the average height of all points within this hemisphere above the disk. This isn't a straightforward average, as we're dealing with a continuous distribution of heights.

The Intuition Behind the Average

Before diving into the math, let's build some intuition. If the hemisphere were a uniform cylinder, the average height would simply be R/2. However, a hemisphere has more volume concentrated near the base and less as we approach the top. This suggests that the average height will be less than R/2. Our calculations will confirm this.

Setting up the Problem

To find the average height, we need to integrate the height function over the area of the disk and then divide by the area of the disk. This essentially sums up all the heights and divides by the total number of points (or, in this continuous case, the area over which the heights are defined).

1. Define the Hemisphere: The equation of a sphere centered at the origin with radius R is x² + y² + z² = R². Since we are only considering the hemisphere above the disk (z ≥ 0), we can express the height z as a function of x and y:

   z = √(R² - x² - y²)

2. Define the Disk: The disk is defined by x² + y² ≤ R² in the xy-plane.

3. The Average Height Formula: The average height, h̄, is given by the following double integral:

   h̄ = (1/A) ∫∫ z dA

   where:

   • A is the area of the disk.
   • z = √(R² - x² - y²) is the height function.
   • dA is the infinitesimal area element.

The Power of Polar Coordinates

The integral is much easier to evaluate using polar coordinates. Here's why:

• Symmetry: The hemisphere and disk are symmetric about the z-axis. Polar coordinates naturally capture this symmetry.
• Simplification: The expression x² + y² transforms to r², simplifying the square root in the height function.
• Natural Integration Limits: The disk is easily described in polar coordinates with r ranging from 0 to R and θ ranging from 0 to 2π.

Transforming to Polar Coordinates

1. Coordinate Transformation:

   • x = r cos θ
   • y = r sin θ
   • dA = r dr dθ

2. Height Function in Polar Coordinates:

   • z = √(R² - r²)

3. Area of the Disk in Polar Coordinates:

   • A = πR²

Setting up the Integral in Polar Coordinates

Substituting these transformations into the average height formula, we get:

h̄ = (1/πR²) ∫₀²π ∫₀ᴿ √(R² - r²) * r dr dθ

Evaluating the Integral: A Step-by-Step Guide

Now, let's tackle the integral step-by-step.

1. Inner Integral (with respect to r):

   • Let u = R² - r²
   • Then du = -2r dr or r dr = -du/2
   • The limits of integration change:
     • When r = 0, u = R²
     • When r = R, u = 0

   The inner integral becomes:

   ∫₀ᴿ √(R² - r²) * r dr = ∫R²⁰ √u * (-du/2) = (1/2) ∫₀R² √u du = (1/2) * (2/3) * u^(3/2) |₀R² = (1/3) * (R²)^(3/2) = (1/3) * R³

2. Outer Integral (with respect to θ):

   The outer integral is now:

   ∫₀²π (1/3) * R³ dθ = (1/3) * R³ ∫₀²π dθ = (1/3) * R³ * θ |₀²π = (1/3) * R³ * (2π) = (2/3) * πR³

3. Calculate the Average Height:

   Finally, substitute the result back into the average height formula:

   h̄ = (1/πR²) * (2/3) * πR³ = (2/3) * R

The Result: A Concise and Elegant Solution

Therefore, the average height of a hemisphere above the disk is (2/3) * R. This confirms our intuition that the average height is less than R/2. The concentration of volume near the base pulls the average height down.

Conceptual Interpretation

This result, h̄ = (2/3)R, can be interpreted geometrically. Imagine a cylinder with height (2/3)R standing on the same disk. The volume of this cylinder represents the "average" volume contained within the hemisphere above the disk.

Exploring Alternative Methods (Optional)

While polar coordinates provide the most straightforward approach, it's possible (though more complex) to solve this problem using Cartesian coordinates directly. This would involve dealing with more complicated integral limits and require careful consideration of the domain of integration. The symmetry inherent in the problem makes polar coordinates the preferred choice.

Applications and Extensions

The concept of finding average values over geometric shapes extends to various fields, including:

• Physics: Calculating the center of mass of an object.
• Engineering: Determining the average stress or strain in a component.
• Computer Graphics: Computing lighting and shading effects.
• Probability and Statistics: Finding expected values of continuous random variables.

The problem of finding the average height of a hemisphere serves as a valuable introduction to these more advanced applications. We can also extend this concept to other shapes like paraboloids or ellipsoids. The core principles of integration and coordinate transformation remain the same, but the specific calculations will vary.

The Importance of Choosing the Right Coordinate System

The solution to this problem highlights the importance of choosing the appropriate coordinate system. While the problem can be formulated in Cartesian coordinates, the resulting integral is significantly more difficult to evaluate. Polar coordinates, by exploiting the symmetry of the hemisphere and disk, simplify the problem dramatically. This is a common theme in calculus and physics: selecting the right coordinate system can often transform a seemingly intractable problem into a manageable one.

Summary of Key Steps

Let's recap the essential steps we took to find the average height:

1. Define the geometry: Clearly define the hemisphere and the disk mathematically.
2. Formulate the integral: Express the average height as a double integral.
3. Transform to polar coordinates: Convert the integral to polar coordinates to simplify the calculations.
4. Evaluate the integral: Carefully evaluate the inner and outer integrals.
5. Interpret the result: Understand the meaning of the final answer.

The Hemisphere in Different Contexts

The hemisphere is a fundamental geometric shape that appears in diverse contexts. From the architectural domes of ancient civilizations to the microscopic structures of biological cells, hemispheres are ubiquitous. Understanding their properties, including average height, is crucial in various scientific and engineering disciplines. For instance, in optics, hemispherical lenses are used to focus light, and the average height influences the lens's focusing power. In fluid dynamics, hemispherical obstacles are used to study flow patterns, and the average height affects the drag force.

Common Mistakes to Avoid

When tackling problems involving integration and geometry, it's crucial to be aware of potential pitfalls:

• Incorrect Integral Limits: Ensure that the limits of integration accurately reflect the region over which you are integrating.
• Forgetting the Jacobian: When transforming coordinates (e.g., from Cartesian to polar), remember to include the Jacobian determinant, which accounts for the change in area or volume. In the case of polar coordinates, dA = r dr dθ.
• Improper Substitution: When using u-substitution, carefully change the limits of integration to match the new variable.
• Sign Errors: Pay close attention to signs during integration and differentiation.

A Word on Numerical Integration

While we were able to solve this problem analytically (i.e., by finding an exact solution), many integrals cannot be solved in closed form. In such cases, numerical integration techniques are employed to approximate the value of the integral. These techniques involve dividing the region of integration into smaller subregions and approximating the integral over each subregion using a numerical method like the trapezoidal rule or Simpson's rule.

Conclusion: A Journey Through Calculus and Geometry

We've successfully navigated the problem of finding the average height of a hemisphere above its disk, employing the powerful tools of calculus and geometry. The journey involved defining the geometry, setting up the integral, transforming to polar coordinates, evaluating the integral, and interpreting the result. The final answer, (2/3)R, provides a concise and elegant solution that highlights the interplay between geometry and integration. This exercise not only provides a concrete answer but also underscores the importance of choosing the right coordinate system and avoiding common integration pitfalls. The concepts explored here are fundamental to various fields, showcasing the versatility and power of calculus in solving real-world problems. The hemisphere, a simple geometric shape, becomes a gateway to understanding more complex phenomena in physics, engineering, and beyond. This exploration demonstrates that even seemingly simple questions can lead to deep and insightful understanding, reinforcing the beauty and elegance of mathematical reasoning.