Problem 65 - Calculus: Early Transcendental Functions

Question

The location \((\bar{x}, \bar{y})\) of the center of gravity (balance point) of a flat plate bounded by \(y=f(x)>0, a \leq x \leq b\) and the \(x\) -axis is given by \(\bar{x}=\frac{\int_{a}^{b} x f(x) d x}{\int_{a}^{b} f(x) d x}\) and \(\bar{y}=\frac{\int_{a}^{b}[f(x)]^{2} d x}{2 \int_{a}^{b} f(x) d x} .\) For the semicircle \(y=f(x)=\sqrt{4-x^{2}},\) use symmetry to argue that \(\bar{x}=0\) and \(\bar{y}=\frac{1}{2 \pi} \int_{0}^{2}\left(4-x^{2}\right) d x .\) Compute \(\bar{y}\)

Step-by-Step Solution

Answer

\(\bar{y} = \frac{16}{3\pi}\)

Step 1: Identify the functions and limits

The semicircle is given by the function \(y = f(x) = \sqrt{4 - x^2}\), with limits \(-2 \leq x \leq 2\) since the radius is 2 units. However, due to symmetry around the y-axis, we only have to compute the integral from 0 to 2 and multiply it by 2.

Step 2: Compute the integral

First, compute the definite integral \(\int_{0}^{2}(4 - x^2)dx\). Multiply the result by 2 to account for semicircle symmetry and then divide it by \(2\pi\) to compute \(\bar{y}\).

Step 3: Compute Final Result

The definite integral \(\int_{0}^{2}(4 - x^2)dx = 8/3\), as calculated using the power rule for integration. Multiply this by 2 and divide by \(2\pi\) to get \(\bar{y} = \frac{16}{3\pi}\).

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus, used to determine the exact area under a curve between two specified points on the x-axis. In the context of finding the center of gravity, we apply the definite integral to calculate areas and weighted averages over a given interval.

• Interval of Integration: The boundaries of the integral in this problem are from 0 to 2, which represent the section of the semicircle being considered from the center (because of symmetry).
• Function Within the Integral: The integral is performed on the function \((4 - x^2)\), which represents the top half (semicircle) of a circle with radius 2. Integrating the function gives us the required area under the curve within the specified limits.
Performing this integration tells us the total "value" from x = 0 to x = 2, a segment that we are analyzing instead of the whole semicircle due to symmetry. Calculating this helps in determining \(\bar{y}\) in our context. The final integral resolves to \(\frac{8}{3}\), as it's computed using another fundamental rule covered in the next section.

Semicircle Symmetry

Symmetry in mathematics often simplifies calculations by reducing the amount of computations that must be performed. In the context of a semicircle centered at the origin along the y-axis, symmetry can serve as a valuable property to consider.

• Horizontal Symmetry: This semicircle is symmetric with respect to the y-axis, meaning that its left half mirrors its right half.
• Center of Gravity: Due to this symmetry, the x-coordinate of the center of gravity \(\bar{x}\) is zero. This implies that any weighted average with respect to the variable x will balance out at the axis.
When symmetry is present, it allows us to compute over just one-half (e.g., from 0 to 2 on the x-axis) and later double the result to account for the entire segment from -2 to 2. Hence, \(\bar{y}\) can be calculated using an integral that covers only one symmetric half of the semicircle, amending our computations with a factor of 2 to reflect the entire shape.

Power Rule for Integration

The power rule is a basic yet powerful tool for finding integrals. It simplifies solving definite integrals of polynomial functions.

• Application of the Rule: The power rule states that the integral of \(x^n\) with respect to x is \(\frac{x^{n+1}}{n+1}\), for all n ≠ -1.
• Usage in the Problem: Here, the function \(4 - x^2\) can be broken down using this rule:
  • The integral of \(4\) is straightforward: \(4x\), as it's akin to \(4x^0\).
  • For \(-x^2\), applying the power rule gives \(-\frac{x^3}{3}\).
Subsequently, applying the endpoints, we substitute x values into \(\frac{8}{3}\) after evaluating at the upper limit and subtracting the result evaluated at the lower limit. These results are then used to calculate our final value for \(\bar{y}\). This simple rule markedly reduces the complexity of solving polynomial integrals.