# Moment of inertia and Change of variables

Using the coordinate change u=xy, v=y/x, set up an iterated integral for the polar moment of inertia of the region bounded by the hyperbola xy=1 , the x-axis, and the two lines x=1 and x=2.

Choose the order of integration which make the limits simplest

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I found something , I just want you to help on it : here is what I have:

the region is attached to the x axis...

so the polar momentum is iqual to:

double integral(y^2)dxdy

---->but y^2=u*v

AND the jacobian matrice of U... and v...

is equal to 2*y/x=2*v

so the inverse is 1/(2v)

THEN WE CAN SAY that

"double integral(y^2)dxdy=(1/(2v))*uv.dvdu" right?

NOW WE FIND THE LIMITS

when x=1 and y=0 we get u=v

when xy=1 we get u=1

finally when x=2 then v=u/4 right?

so we see that we have

double integral(u/2)dudv, for u varying from 1 to v

and v varying from 1 to v, no?

My result then is "1/24(1-v^3)"

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#### Solution Summary

The solution demonstrates how to apply change of variables to a double integral, from the change in the integrand to the change in integration limits.

The solution contains 3 pages of completre derivations.