Multivariable and Vector Calculus · Multiple integration and change of variables
Straighten a product-ratio region with logarithms
Problem
In the first quadrant let \[ D=\{(x,y):1\le xy\le e^2,\ 1\le y/x\le e^3\}. \] Use \[ u=\ln(xy),\qquad v=\ln(y/x) \] to evaluate \[ I=\iint_D\frac{\ln(xy)+\ln(y/x)}{xy}\,dA. \] Derive the inverse formulas for $x$ and $y$, compute the signed and absolute Jacobian, describe the image rectangle with its orientation, and give the exact value of $I$. Explain why restricting to the first quadrant is essential for this coordinate choice.
Hint
Add and subtract the equations for $u$ and $v$.
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