# Homework Help: Use the given transformation to evaluate the integral

1. Jul 17, 2012

### 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.

2. Jul 17, 2012

### SammyS

Staff Emeritus
That looks good to me.

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

Try the integration.

3. Jul 17, 2012

### ptguard1

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