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Chapter 11: Problem 79

Find the centroid of the region that is bounded below by the \(x\) -axis and above by the ellipse \(\left(x^{2} / 9\right)+\left(y^{2} / 16\right)=1\)

Short Answer

Expert verified
The centroid is at \((0, \frac{16}{3\pi})\).

Step by step solution

01

Understanding the Problem

The problem asks us to find the centroid of the region bounded by the x-axis and an ellipse given by \(\frac{x^2}{9} + \frac{y^2}{16} = 1\). This ellipse is centered at the origin with a semi-major axis of 4 units along the y-axis and a semi-minor axis of 3 units along the x-axis.
02

Establishing the Boundaries

The region of interest is bounded by the x-axis, or \(y = 0\), and the part of the ellipse above this axis, where \(y > 0\). The boundaries for integration in the x-direction are from \(-3\) to \(3\), because these are the x-intercepts of the ellipse.
03

Finding the Area of the Region

The area \(A\) of the region bounded by the x-axis and the top half of the ellipse can be determined by integrating the ellipse's equation with respect to \(x\). Specifically, calculate \(A = 2 \int_{-3}^{3} \frac{4}{3} \sqrt{9 - x^2} \, dx\) using the formula for \(y\) derived from the ellipse equation. Using a trigonometric substitution or known ellipse area calculation, this results in \(A = \frac{3\pi}{2}\).
04

Calculating the x-coordinate of the Centroid

The x-coordinate of the centroid, \(\bar{x}\), is found using symmetry as \(\bar{x} = 0\) because the region is symmetric about the y-axis.
05

Calculating the y-coordinate of the Centroid

The y-coordinate of the centroid \(\bar{y}\) is calculated using \(\bar{y} = \frac{1}{A} \int_{-3}^{3} \frac{4}{3} \sqrt{9 - x^2} \cdot \frac{4}{3}\sqrt{9 - x^2} \, dx\). Evaluating this integral and simplifying gives \(\bar{y} = \frac{16}{3\pi}\).
06

Conclusion

The centroid of the region is found at the point \((0, \frac{16}{3\pi})\). This result places the centroid on the y-axis above the x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ellipse Equation
An ellipse is a shape that looks like a stretched circle. Its equation in the standard form is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively. In this case, the ellipse given is \(\frac{x^2}{9} + \frac{y^2}{16} = 1\). Here, \(a = 3\) and \(b = 4\), indicating the semi-major axis is along the y-axis due to 16 being bigger than 9.

**Key Characteristics of an Ellipse:**
  • **Center at Origin**: The given equation implies the ellipse is centered at the origin, point (0,0).
  • **Axes**: The longest axis is vertical (along the y-axis) because 16 is greater than 9, making \(b\) the semi-major axis length.
  • **Intercepts**: The ellipse intersects the x-axis at points (-3,0) and (3,0), and the y-axis at points (0,-4) and (0,4).
Understanding these properties helps to visualize and analyze the ellipse, especially when determining regions and applying further mathematical methods like integration.
Integration
Integration is a fundamental technique used in calculus to calculate areas under curves, among other applications. In this exercise, we use integration to find the area of the region above the x-axis bounded by the ellipse. This area is crucial as it is required for calculating the centroid.

**Steps in Integration Here:**
  • Set up the integral to find the area of the top half of the ellipse. This involves integrating the function derived from the ellipse equation for \(y\), specifically \(y = \frac{4}{3} \sqrt{9 - x^2}\).
  • The integration bounds are from \(-3\) to \(3\), the x-intercepts of the ellipse.
  • Using trigonometric substitution or known ellipse area formulas, this integral evaluates to an area \(A = \frac{3\pi}{2}\).
This calculated area is not just a number, but a crucial figure used in the computing of the y-coordinate of the centroid, making integration a vital tool in geometry and calculus.
Symmetry in Geometry
Symmetry plays a significant role in simplifying problems involving geometric shapes like ellipses. Symmetry can drastically reduce the complexity of computations, especially in determining centroids.

**Why Symmetry Matters:**
  • The given region is symmetric about the y-axis, meaning if you fold it along this axis, both halves would match.
  • Due to this symmetry, the x-coordinate of the centroid, \(\bar{x}\), is immediately determined as 0, without requiring complex calculations.
This inherent symmetry simplifies our task, as it allows us to focus on calculating only the y-component, reducing the overall effort needed to find centroid coordinates.
Centroid Calculation
Calculating the centroid involves figuring out what is effectively the 'center of mass' for a geometric region. For the region bounded by the x-axis and the ellipse here, the centroid is determined through mathematical formulas and simplifications.

**Finding the Centroid's Coordinates:**
  • **x-coordinate (\(\bar{x}\))**: Because of symmetry about the y-axis, \(\bar{x} = 0\).
  • **y-coordinate (\(\bar{y}\))**: It requires calculation by integrating the function \((\frac{4}{3} \sqrt{9 - x^2})^2\) and dividing by the area \(A\).
    Simultaneously, this translates to:
    \(\bar{y} = \frac{1}{A} \int_{-3}^{3} \left(\frac{4}{3} \sqrt{9 - x^2}\right)^2 dx = \frac{16}{3\pi}\).
This leads us to conclude that the centroid is located at the point \((0, \frac{16}{3\pi})\), clearly emphasizing the importance of both mathematical symmetry and accuracy during the computational process.

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