A) Use Green's theorem to compute the area inside the = 1. 2² y² ellipse + 19² 14² Use the fact that the area can be written as
A) Use Green's theorem to compute the area inside the = 1. 2² y² ellipse + 19² 14² Use the fact that the area can be written as
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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![**Transcription for Educational Website:**
---
### Topic: Calculating Areas Using Green's Theorem and Parametrization
**A)**
Use Green's theorem to compute the area inside the ellipse defined by:
\[
\frac{x^2}{19^2} + \frac{y^2}{14^2} = 1.
\]
Green's theorem states that the area can be represented as:
\[
\iint_D dx \, dy = \frac{1}{2} \oint_{\partial D} -y \, dx + x \, dy.
\]
**Hint:** Use the parametrization \(x(t) = 19 \cos(t)\).
**Result:** The area of the ellipse is **836**.
---
**B)**
To find the area of the interior defined by the curve:
\[
x^{2/3} + y^{2/3} = 9^{2/3},
\]
find a suitable parametrization of the curve.
**Hint:** Use the parametrization \(x(t) = 9 \cos^3(t)\).
**Result:** The area is \(\frac{81\pi}{4}\).
---
These exercises illustrate techniques for calculating areas enclosed by curves using calculus tools like Green's Theorem and curve parametrization.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff35f8f89-65fc-4b90-98e9-68f8eaa1b3da%2F09a168c3-f873-4164-b2be-02003b75d9f1%2Ftjxfznd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
---
### Topic: Calculating Areas Using Green's Theorem and Parametrization
**A)**
Use Green's theorem to compute the area inside the ellipse defined by:
\[
\frac{x^2}{19^2} + \frac{y^2}{14^2} = 1.
\]
Green's theorem states that the area can be represented as:
\[
\iint_D dx \, dy = \frac{1}{2} \oint_{\partial D} -y \, dx + x \, dy.
\]
**Hint:** Use the parametrization \(x(t) = 19 \cos(t)\).
**Result:** The area of the ellipse is **836**.
---
**B)**
To find the area of the interior defined by the curve:
\[
x^{2/3} + y^{2/3} = 9^{2/3},
\]
find a suitable parametrization of the curve.
**Hint:** Use the parametrization \(x(t) = 9 \cos^3(t)\).
**Result:** The area is \(\frac{81\pi}{4}\).
---
These exercises illustrate techniques for calculating areas enclosed by curves using calculus tools like Green's Theorem and curve parametrization.
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