Approximate the area of the graph of the ellipse 4x² +9y² = 36
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Title: Calculating the Area of an Ellipse**
**Objective:**
To approximate the area of the ellipse given by the equation:
\[ 4x^2 + 9y^2 = 36 \]
**Introduction:**
In mathematics, the area of an ellipse is a fundamental concept, often encountered in geometry and calculus. An ellipse is defined by its equation in the form of \( Ax^2 + By^2 = C \).
**Clarification:**
1. **Standard Form Conversion**:
The given ellipse equation is:
\[ 4x^2 + 9y^2 = 36 \]
First, rewrite it in the standard form by dividing the entire equation by 36:
\[
\frac{x^2}{9} + \frac{y^2}{4} = 1
\]
From this, we identify:
- Semi-major axis \( a = 3 \)
- Semi-minor axis \( b = 2 \)
2. **Area Calculation**:
The area \( A \) of an ellipse is given by the formula:
\[ A = \pi ab \]
Substituting the values of \( a \) and \( b \):
\[ A = \pi \times 3 \times 2 = 6\pi \]
Therefore, the approximate area of the ellipse is \( 6\pi \) square units.
**Conclusion:**
Understanding how to manipulate the equation of an ellipse into its standard form is crucial for solving many problems related to ellipses, including finding their areas. The process demonstrated simplifies these calculations, making it a valuable tool in mathematical problem-solving.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2f2c4bd-bf4c-4a3b-a0a2-6333c3306a45%2F1dc1cfd2-52bb-4f49-801e-0edab3adda0c%2Fkil30zs_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Area of an Ellipse**
**Objective:**
To approximate the area of the ellipse given by the equation:
\[ 4x^2 + 9y^2 = 36 \]
**Introduction:**
In mathematics, the area of an ellipse is a fundamental concept, often encountered in geometry and calculus. An ellipse is defined by its equation in the form of \( Ax^2 + By^2 = C \).
**Clarification:**
1. **Standard Form Conversion**:
The given ellipse equation is:
\[ 4x^2 + 9y^2 = 36 \]
First, rewrite it in the standard form by dividing the entire equation by 36:
\[
\frac{x^2}{9} + \frac{y^2}{4} = 1
\]
From this, we identify:
- Semi-major axis \( a = 3 \)
- Semi-minor axis \( b = 2 \)
2. **Area Calculation**:
The area \( A \) of an ellipse is given by the formula:
\[ A = \pi ab \]
Substituting the values of \( a \) and \( b \):
\[ A = \pi \times 3 \times 2 = 6\pi \]
Therefore, the approximate area of the ellipse is \( 6\pi \) square units.
**Conclusion:**
Understanding how to manipulate the equation of an ellipse into its standard form is crucial for solving many problems related to ellipses, including finding their areas. The process demonstrated simplifies these calculations, making it a valuable tool in mathematical problem-solving.
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