Problem 289 - Calculus Volume 1

Question

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area \(M\) and the centroid \((\overline{x}, \overline{y})\) for the given shapes. Use symmetry to help locate the center of mass whenever possible. Half-ring: \(\quad y^{2}+x^{2}=1, \quad y^{2}+x^{2}=4, \quad\) and \(y=0\)

Step-by-Step Solution

Answer

The area is \(\frac{3}{2}\pi\), and the centroid is \((0, \frac{1}{3})\)."

Step 1: Understanding the Shape

We are given two circles with equations \(y^2 + x^2 = 1\) and \(y^2 + x^2 = 4\), along with the line \(y = 0\). The shapes describe two concentric circles. We are instructed to consider the upper halves of these circles due to the line \(y = 0\).

Step 2: Calculate the Area of the Region

The region enclosed is the area between the two semicircles. The area of a full circle is \(\pi r^2\). Thus, the area of the smaller circle (radius 1) is \(\pi\), and the area of the larger circle (radius 2) is \(4\pi\). The upper halves will have half these areas, i.e., \(\frac{1}{2}\pi\) and \(2\pi\) respectively. The area of the half-ring region is the difference: \[ M = 2\pi - \frac{1}{2}\pi = \frac{3}{2}\pi.\]

Step 3: Analyze Symmetry

Using symmetry, we note that because both semicircles and the line about \(y = 0\) are symmetrically distributed about the \(y\)-axis, the \(\overline{x}\) component of the centroid will be 0. This is because the shapes are evenly distributed to the left and right of the origin.

Step 4: Calculate \(\overline{y}\) Using Centroid Formula

The formula for the centroid \(\overline{y}\) of a curved line is more complex, but for symmetry in semi-circular areas:\[ \overline{y} = \frac{2}{3} \text{height of semicircle}. \]Applying this to our ring, analyze the separate semicircles. For the smaller semicircle, height is 1, for the larger, it's 2. Overall,\(\overline{y}\) lies halfway between the semicircles, meaning it's:\[ \overline{y} = \frac{\frac{2}{3} \cdot 1 + \frac{2}{3} \cdot 2}{2} = \frac{1}{3}.\]

Step 5: Compile Final Results

Summarizing, the area \(M\) of the half-ring is \(\frac{3}{2}\pi\), and the centroid is located at \((\overline{x}, \overline{y}) = (0, \frac{1}{3})\).

Key Concepts

Area of Semicircle

Understanding how to find the area of a semicircle is vital when dealing with problems involving partially enclosed circular shapes. A semicircle is simply half of a circle. To calculate its area, you start with the formula for the area of a full circle, which is \( \pi r^2 \). Since a semicircle is half of the circle, the area of a semicircle is given by:

• \( A_{\text{semicircle}} = \frac{1}{2} \pi r^2 \)

In the given problem, we are dealing with two semicircles of different radii. The smaller semicircle has a radius of 1, yielding an area of \( \frac{1}{2} \pi \). The larger semicircle, with a radius of 2, has an area of \( 2\pi \). The total area of the semicircular ring is found by subtracting the area of the smaller semicircle from the larger: \( M = 2\pi - \frac{1}{2}\pi = \frac{3}{2}\pi \).This step is crucial for finding the solution to this exercise. By understanding how to compute these basic areas, you can tackle more complex geometric problems effectively.

Centroid Calculation

The centroid, or geometric center, of a shape is a point where it can be balanced perfectly. Calculating the centroid of complex shapes, like the semicircular region in this problem, starts with understanding the properties of its individual parts.
For semicircular shapes, symmetry plays an essential role in simplifying these calculations.The problem uses the formula for the centroid of a semicircle, which is a critical factor in physics and engineering where it refers to the center of mass of an object.

• For a semicircle, the vertical centroid point \( \overline{y} \) is \( \frac{2}{3} \) of its radius \( r \) from the base of the semicircle.

Since the traditional x-axis (horizontal symmetry line) is also the line of symmetry here, the \( \overline{x} \) value is 0.
In the given half-ring, the average \( \overline{y} \) is calculated across both semicircle areas: \[ \overline{y} = \frac{\frac{2}{3} \cdot 1 + \frac{2}{3} \cdot 2}{2} = \frac{1}{3} \]This method of calculating centers uses the properties of symmetry and balance, allowing quick determination of centroids without exhaustive calculations.

Symmetry in Geometry

Symmetry is a powerful concept in geometry that simplifies complex problems. Geometric symmetry involves seeing how shapes can be divided into parts that mirror each other.
In the context of this problem, symmetry is seen with the semicircles placed around the origin along the horizontal axis \( y=0 \).Here's why symmetry is helpful:

• It allows simplification of calculations, by reducing the need for manual computations along the symmetry line.
• For identical parts on either side of a line of symmetry, certain properties like centroids can be deduced instantly."

In this exercise, by observing the symmetry about the \( y \)-axis, the \( \overline{x} \) component of the centroid was directly established as 0. This reflects the concept's applicative nature: understanding symmetry not only simplifies calculations but also deepens comprehension of geometry's aesthetic balance.