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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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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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.
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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