6. Approximate the area of the graph of the ellipse 4x² +9y² n=2 = 36 using Gaussian quadrature with
6. Approximate the area of the graph of the ellipse 4x² +9y² n=2 = 36 using Gaussian quadrature with
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 6:** Approximate the area of the graph of the ellipse \(4x^2 + 9y^2 = 36\) using Gaussian quadrature with \(n=2\).
**Explanation:**
This problem involves finding the area enclosed by the given ellipse using Gaussian quadrature, a numerical integration method. Here, \(n=2\) indicates using two points for the Gaussian quadrature, which provides an efficient way to approximate the integral representing the area of the ellipse.
The equation \(4x^2 + 9y^2 = 36\) represents an ellipse, which can also be rewritten in standard form to facilitate the calculation:
\[
\frac{x^2}{9} + \frac{y^2}{4} = 1
\]
The semi-major and semi-minor axes can be identified from this form:
- Semi-major axis: \(a = 3\)
- Semi-minor axis: \(b = 2\)
The area \(A\) of an ellipse is given by the formula:
\[
A = \pi \cdot a \cdot b = \pi \cdot 3 \cdot 2 = 6\pi
\]
Gaussian quadrature will provide an approximation for this integral, depending on the function representation and the selection of appropriate weights and nodes for \(n=2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2f2c4bd-bf4c-4a3b-a0a2-6333c3306a45%2F915d80cf-e879-42b2-b1b5-b0db324a40a4%2Fpkmbjki_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 6:** Approximate the area of the graph of the ellipse \(4x^2 + 9y^2 = 36\) using Gaussian quadrature with \(n=2\).
**Explanation:**
This problem involves finding the area enclosed by the given ellipse using Gaussian quadrature, a numerical integration method. Here, \(n=2\) indicates using two points for the Gaussian quadrature, which provides an efficient way to approximate the integral representing the area of the ellipse.
The equation \(4x^2 + 9y^2 = 36\) represents an ellipse, which can also be rewritten in standard form to facilitate the calculation:
\[
\frac{x^2}{9} + \frac{y^2}{4} = 1
\]
The semi-major and semi-minor axes can be identified from this form:
- Semi-major axis: \(a = 3\)
- Semi-minor axis: \(b = 2\)
The area \(A\) of an ellipse is given by the formula:
\[
A = \pi \cdot a \cdot b = \pi \cdot 3 \cdot 2 = 6\pi
\]
Gaussian quadrature will provide an approximation for this integral, depending on the function representation and the selection of appropriate weights and nodes for \(n=2\).
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