Region D of the (x, y)-plane is the locus of points satisfying the conditions x > 0, −1 – log x ≤ y ≤ – log x and 2 log x ≤ y ≤ 2+2 log x. (a) Sketch and identify region D in the (x, y)-plane. (b) Calculate the Jacobian of the transformation x=u1/3√-1/3 y=log(u²/31/3), where log indicates a natural logarithm (to base e). You may find it easier to re-write the formula for y before differentiating it. (c) Hence evaluate the integral x² dx dy. SS x² D Hint: Begin by expressing the boundaries of D in terms of transformed variables.
Region D of the (x, y)-plane is the locus of points satisfying the conditions x > 0, −1 – log x ≤ y ≤ – log x and 2 log x ≤ y ≤ 2+2 log x. (a) Sketch and identify region D in the (x, y)-plane. (b) Calculate the Jacobian of the transformation x=u1/3√-1/3 y=log(u²/31/3), where log indicates a natural logarithm (to base e). You may find it easier to re-write the formula for y before differentiating it. (c) Hence evaluate the integral x² dx dy. SS x² D Hint: Begin by expressing the boundaries of D in terms of transformed variables.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 77E
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Transcribed Image Text:Region D of the (x, y)-plane is the locus of points satisfying the conditions x > 0,
−1 – log x ≤ y ≤ – log x and 2 log x ≤ y ≤ 2+2 log x.
(a) Sketch and identify region D in the (x, y)-plane.
(b) Calculate the Jacobian of the transformation
x=u1/3√-1/3
y=log(u²/31/3),
where log indicates a natural logarithm (to base e). You may find it easier to re-write the
formula for y before differentiating it.
(c) Hence evaluate the integral
x² dx dy.
SS x²
D
Hint: Begin by expressing the boundaries of D in terms of transformed variables.
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