Problem 284 - Calculus Volume 1

Question

For the following exercises, use the theorem of Pappus to determine the volume of the shape. A sphere created by rotating a semicircle with radius \(a\) around the \(y\) -axis. Does your answer agree with the volume of a sphere?

Step-by-Step Solution

Answer

Yes, the answer agrees with the volume of a sphere, \( \frac{4}{3} \pi a^3 \).

Step 1: Understanding Pappus's Theorem

Pappus's Theorem states that the volume of a solid of revolution generated by revolving a plane curve about an external axis is equal to the product of the area of the curve and the distance traveled by the centroid of the curve during the revolution.

Step 2: Define the Planar Shape and Its Area

The shape before revolving is a semicircle with radius \( a \). The area of a semicircle is given by \( \frac{1}{2} \pi a^2 \).

Step 3: Determine the Centroid Location

For a semicircle with radius \( a \) that is symmetric around the y-axis, the centroid lies on the y-axis at a distance of \( \frac{4a}{3\pi} \) from the center of the circle.

Step 4: Calculate the Distance Traveled by the Centroid

When the semicircle is rotated around the y-axis, the centroid travels in a circular path with a radius equal to its x-coordinate, \( \frac{4a}{3\pi} \). The circumference of this path is \( 2\pi \times \frac{4a}{3\pi} = \frac{8a}{3} \).

Step 5: Apply Pappus's Theorem

According to Pappus's Theorem, the volume of the solid (a sphere in this case) is equal to the area of the semicircle multiplied by the distance traveled by its centroid. Substitute the values: \( V = \frac{1}{2} \pi a^2 \times \frac{8a}{3} = \frac{4}{3} \pi a^3 \).

Step 6: Verification with Sphere Volume Formula

The calculated volume \( \frac{4}{3} \pi a^3 \) using Pappus's Theorem is the well-known formula for the volume of a sphere with radius \( a \). Thus, the calculation is consistent with the known formula for the volume of a sphere.

Key Concepts

Volume of Revolution

To grasp the concept of the "volume of revolution," imagine spinning a 2D shape around an axis to create a 3D object. This is what happens when a shape, like a semicircle, is rotated. The volume of the 3D object produced can be calculated using mathematical principles.

Pappus's Theorem offers a useful shortcut for finding this volume by relating it to characteristics of the 2D shape. According to this theorem, the volume is the product of the area of the shape being revolved and the distance the centroid of that shape travels during the rotation.

In this exercise, a semicircle is rotated around the y-axis. By using Pappus's Theorem, you can calculate the volume of the resulting sphere, providing a clear understanding of how shapes transform through rotation.

Centroid Calculation

Centroid calculation is crucial for using Pappus's Theorem. The centroid is like the "center of gravity" for a shape, and its location is determined by its geometry.

For a semicircle, the centroid isn't at the exact geometric center because half the circle is missing. Instead, it's slightly up from what would be the center of a full circle. In mathematical terms, this point for a semicircle with radius \( a \) is located at a distance of \( \frac{4a}{3\pi} \) along the y-axis from the center.

Once the centroid is accurately determined, it's used to find the path it traces when the semicircle rotates around the y-axis. This path is crucial for applying Pappus's Theorem, as the distance traveled by the centroid is part of the formula for calculating the volume of revolution.

Semicircle Area

The area of a semicircle is half that of a complete circle, which can be helpful when calculating the volume of a solid of revolution.

For a full circle, the area is \( \pi a^2 \), where \( a \) is the radius. Consequently, the area of a semicircle becomes \( \frac{1}{2} \pi a^2 \). This reduction is crucial because it factors into Pappus's Theorem.

To find the volume of the sphere created by rotating the semicircle, the area must be multiplied by the distance the centroid travels. This area calculation acts as the foundation for the entire process, contributing to the understanding and application of the theorem effectively.

Sphere Volume Calculation

The calculation of sphere volume through geometric principles confirms the result obtained using Pappus's Theorem.

Usually, the volume of a sphere is found using the formula \( \frac{4}{3} \pi a^3 \), where \( a \) is the radius of the sphere. This formula represents the uniform space contained within the sphere.

In this exercise, using Pappus's Theorem, you also arrive at this exact sphere volume. By multiplying the semicircle's area by the distance the centroid travels, we got \( \frac{4}{3} \pi a^3 \), which perfectly matches the well-known sphere volume formula.

This consistency provides a great way to check the validity of our solution and demonstrates the effectiveness of Pappus's Theorem in practical scenarios involving volume projections.