Vector Calculus | Part IA, 2001

State the rule for changing variables in a double integral.

Let DD be the region defined by

{1/xy4x when 12x1xy4/x when 1x2\begin{cases}1 / x \leq y \leq 4 x & \text { when } \frac{1}{2} \leq x \leq 1 \\ x \leq y \leq 4 / x & \text { when } 1 \leq x \leq 2\end{cases}

Using the transformation u=y/xu=y / x and v=xyv=x y, show that

D4xy3x2+y2dxdy=152ln172\int_{D} \frac{4 x y^{3}}{x^{2}+y^{2}} d x d y=\frac{15}{2} \ln \frac{17}{2}

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