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In order to find the area of an ellipse, we may make use of the idea of transformations and our knowledge of (x - 5)² (y+4)² the area of a circle. For example, consider R, the region bounded by the ellipse 16 1 The easiest transformation to choose makes U= which should be easily inverted to obtain x = leading to a Jacobian of a (x, y) a(u, v) and v= = and y = + a (x, y) And since · SS₂₁A = - dudu where the transformed region S is bounded by u²+ v² = 1, we - J1₁ s d(u, v) calculate the area by multiplying the area and the Jacobian, arriving at (give an exact answer) = 1.

In order to find the area of an ellipse, we may make use of the idea of transformations and our knowledge of (x - 5)² (y+4)² the area of a circle. For example, consider R, the region bounded by the ellipse 16 1 The easiest transformation to choose makes U= which should be easily inverted to obtain x = leading to a Jacobian of a (x, y) a(u, v) and v= = and y = + a (x, y) And since · SS₂₁A = - dudu where the transformed region S is bounded by u²+ v² = 1, we - J1₁ s d(u, v) calculate the area by multiplying the area and the Jacobian, arriving at (give an exact answer) = 1.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.2: Ellipses
Problem 51E
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In order to find the area of an ellipse, we may make use of the idea of transformations and our knowledge of (x - 5)² (y + 4)² the area of a circle. For example, consider R, the region bounded by the ellipse + 16 1 The easiest transformation to choose makes Ա — which should be easily inverted to obtain I= leading to a Jacobian of a (x, y) a(u, v) And since and v= = and y= a(x, y) ₁² - dudu where the transformed region S is bounded by u²+ v² = 1, we a(u, v) dA= calculate the area by multiplying the area and the Jacobian, arriving at (give an exact answer) = 1.
Transcribed Image Text:In order to find the area of an ellipse, we may make use of the idea of transformations and our knowledge of (x - 5)² (y + 4)² the area of a circle. For example, consider R, the region bounded by the ellipse + 16 1 The easiest transformation to choose makes Ա — which should be easily inverted to obtain I= leading to a Jacobian of a (x, y) a(u, v) And since and v= = and y= a(x, y) ₁² - dudu where the transformed region S is bounded by u²+ v² = 1, we a(u, v) dA= calculate the area by multiplying the area and the Jacobian, arriving at (give an exact answer) = 1.
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