2 Attachment(s)

Transformations to evaluate an integral

Attachment 28785

The book prescribes the transformation T(*u,v*) = (*u*^{2} - *v*^{2}, 2*u**v*) where x = *u*^{2} - *v*^{2} and y = 2*u**v*. Using this information, I have found that the Jacobian of the transformation as 4(*u*^{2}* + v*^{2}), resulting in the following:

Attachment 28786

I am just unable to find *R*, the region of integration. I graphed it in the *xy* plane looked like a the tip of a leaf protruding over the x-axis.

My biggest problem is that this *is not* a linear transformation and my professor hasn't taught us how to tackle those.

Any help will be greatly appreciated,

--Giest

Re: Transformations to evaluate an integral

You don't need to do a transformation at all...

If you draw the region and picture drawing horizontal strips, these strips are bounded on the left by and bounded on the right by , and then these strips are summed up between and . So your integral is

Re: Transformations to evaluate an integral

That thought certainly occurred to me, however, the course I need this for is a Linear Algebra course, so I am pretty sure my professor wants us to use linear transformations. I'll definitely use this to check my answer, though.

Re: Transformations to evaluate an integral

If you must use this transformation, when and , then that means your bounding functions become

and

So your boundary is a square with u and v both bounded by . Can you see what the bounds for your region must be now?

Re: Transformations to evaluate an integral

Ah, Thanks! Now that you have shown me, it's shamefully simple. I was just trying to map the vertices of the region from the xy plane to the uv plane, when, in stead, I should have been mapping the entire function.

Thanks, again!