T Let D be an ellipse rotated counterclockwise by radians, as pictured below. Before rotation, the 6 horizontal radius was 6 and vertical radius was 3. Convert t (8x + 4y)dA into an integral over a polar integral over a unit circle. (type theta for 6) T 2x dr de
T Let D be an ellipse rotated counterclockwise by radians, as pictured below. Before rotation, the 6 horizontal radius was 6 and vertical radius was 3. Convert t (8x + 4y)dA into an integral over a polar integral over a unit circle. (type theta for 6) T 2x dr de
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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![The problem involves an ellipse \( D \) that is rotated counterclockwise by \( \frac{\pi}{6} \) radians. Initially, the horizontal radius of the ellipse was 6 and the vertical radius was 3. The illustration shows the ellipse on a Cartesian coordinate system, rotated around the origin.
The task is to convert the double integral \(\iint_D (8x + 4y) \, dA \) into an integral over a polar coordinate system for a unit circle. This involves changing the Cartesian coordinates to polar coordinates, where the conversion formulas are \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
The expression for the double integral in polar coordinates is given as:
\[
\int_0^{2\pi} \int_0^1 \qquad dr \, d\theta
\]
To solve this, you need to apply these conversions and adjust the limits of integration appropriately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20dfe5a-a4c1-4793-9f05-a80ad59a67d4%2Ff7fbbc4b-2341-40c6-a8e7-c5971c004cc6%2Furs5k9k_processed.png&w=3840&q=75)
Transcribed Image Text:The problem involves an ellipse \( D \) that is rotated counterclockwise by \( \frac{\pi}{6} \) radians. Initially, the horizontal radius of the ellipse was 6 and the vertical radius was 3. The illustration shows the ellipse on a Cartesian coordinate system, rotated around the origin.
The task is to convert the double integral \(\iint_D (8x + 4y) \, dA \) into an integral over a polar coordinate system for a unit circle. This involves changing the Cartesian coordinates to polar coordinates, where the conversion formulas are \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
The expression for the double integral in polar coordinates is given as:
\[
\int_0^{2\pi} \int_0^1 \qquad dr \, d\theta
\]
To solve this, you need to apply these conversions and adjust the limits of integration appropriately.
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