Use the given transformation to evaluate the integral

[ptguard1]

∫∫10xy(dA), where R is the region in the first quadrant bounded by the lines y=x/2 and y=2x/3 and by the hyperbolas xy=1/2 and xy=3/2

The transformations given in the problem (these cannot be altered): x=u/v and y=v
Relevant equations:

The Jacobian - ∂(x,y)/∂(u,v)
The attempt at a solution:

y=(3/2)x: 2v^2=3u
y=x/2: 2v^2=u
xy=1/2: u=1/2
xy=3/2: u=3/2

After making the transformations, I get the following double integral:

10∫(u from 1/2 to 3/2)∫(v from √(u/2) to √(3u/2)) (u/v)dvdu

I feel like my transformations are suppose to result in basic bounds without variables, so I think I am doing this problem incorrectly and can't figure out any other way to go about it.

 

[SammyS]

∫∫10xy(dA), where R is the region in the first quadrant bounded by the lines y=x/2 and y=2x/3 and by the hyperbolas xy=1/2 and xy=3/2

The transformations given in the problem (these cannot be altered): x=u/v and y=v

Relevant equations:

The Jacobian - ∂(x,y)/∂(u,v)

The attempt at a solution:

y=(3/2)x: 2v^2=3u
y=x/2: 2v^2=u
xy=1/2: u=1/2
xy=3/2: u=3/2

After making the transformations, I get the following double integral:

10∫(u from 1/2 to 3/2) ∫(v from √(u/2) to √(3u/2)) (u/v)dvdu

I feel like my transformations are suppose to result in basic bounds without variables, so I think I am doing this problem incorrectly and can't figure out any other way to go about it.

That looks good to me.

The transformation does simplify the region, even if it's not rectangular.

Try the integration.

 

[ptguard1]

Wonderful! I performed the integration and got 10ln(√3) and this was correct.