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Calculus / Calculus: Early Transcendentals 1 / Chapter 13.7 / Problem 51E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Solution: Ellipse problems Let R be the region bounded by

Chapter 14, Problem 51E

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QUESTION:

Ellipse problems Let R be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\), where a > 0 > and b > 0 are real numbers. Let T be the transformation x = au, y = bv.

Find the average square of the distance between points of R and the origin..

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QUESTION:

Ellipse problems Let R be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\), where a > 0 > and b > 0 are real numbers. Let T be the transformation x = au, y = bv.

Find the average square of the distance between points of R and the origin..

ANSWER:

Step 1 of 4

Let R be the region bounded by the ellipse , where $a>0,\ b>0$ are real numbers.

Let T be the transformation $x=au,\ y=bv$ .

Let us find the average square of the distance between points of R and the origin.

Let us find out the region the ellipse is transformed in the $uv$ plane using the given transformation.

We are given that $x=au,\ y=bv$ .

Therefore the ellipse becomes

Therefore we can conclude that the ellipse becomes a unit circle in the $uv$ plane, the unit circle with area $\pi{}$

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