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- Jan 26, 2012

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Thanks to those who participated in last week's POTW. I'll just say it's no fun unless more people participate!

I got some complaints about the difficulty of some of the previous problems; so, for this week's problem, I've decided to go with something a "tad" easier and doable -- a problem from multivariable calculus.

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\[\iint\limits_R \sin(xy)\,dA\]

by making an appropriate change of variables where $R$ is the region enclosed by the curves $xy=\pi$, $xy=2\pi$, $xy^4=1$ and $xy^4=2$.

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I will provide a hint/suggestion for this week's problem:

Use the fact that if $\displaystyle\frac{\partial(x,y)}{\partial(u,v)}= \begin{vmatrix} \frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{vmatrix}$ is the Jacobian of our transformation, then $\dfrac{\partial(x,y)}{\partial(u,v)} = \dfrac{1}{\partial(u,v)/\partial(x,y)}$.

This form of the Jacobian comes in handy when we can't explicitly solve for $x$ and $y$ in terms of $u$ and $v$.

Remember to read the POTW submission guidlines to find out how to submit your answers!

I got some complaints about the difficulty of some of the previous problems; so, for this week's problem, I've decided to go with something a "tad" easier and doable -- a problem from multivariable calculus.

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**Problem**: Evaluate\[\iint\limits_R \sin(xy)\,dA\]

by making an appropriate change of variables where $R$ is the region enclosed by the curves $xy=\pi$, $xy=2\pi$, $xy^4=1$ and $xy^4=2$.

-----

I will provide a hint/suggestion for this week's problem:

This form of the Jacobian comes in handy when we can't explicitly solve for $x$ and $y$ in terms of $u$ and $v$.

Remember to read the POTW submission guidlines to find out how to submit your answers!

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