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1. 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 logx. (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(u²/³v1/3), 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 SS x² dx dy. D Hint: Begin by expressing the boundaries of D in terms of transformed variables.

1. 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 logx. (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(u²/³v1/3), 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 SS x² dx dy. D Hint: Begin by expressing the boundaries of D in terms of transformed variables.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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1. 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 logx. (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(u²/³v1/3), 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 SS x² dx dy. D Hint: Begin by expressing the boundaries of D in terms of transformed variables.
Transcribed Image Text:1. 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 logx. (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(u²/³v1/3), 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 SS x² dx dy. D Hint: Begin by expressing the boundaries of D in terms of transformed variables.
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