Question

The shaded region is bounded by curves x=1, y=1+x, and y=2(1+ 1/x), consider the transformation defined by x = u/v and y= u+v. What are u(x,y), v(x,y), x=1=>, y=1+x=>, and y=2(1+ 1/x)=>?

Answer 1

solve for u and v using substitution:
\[u = x*v\]
thus
\[y = x*v +v\]
\[y = v(x+1)\]
\[v = \frac{y}{x+1}\]
plug back in to solve for u:
\[u = \frac{xy}{x+1}\]
Now transform the "x,y" equations into "u,v" form
\[x=1 \rightarrow \frac{u}{v}=1\rightarrow v=u\]
\[y=x+1 \rightarrow v = \frac{x+1}{x+1} \rightarrow v=1\]
Note:
\[y =2(1+\frac{1}{x}) \rightarrow y = \frac{2(x+1)}{x}\]
\[\rightarrow u = \frac{x*\frac{2(x+1)}{x}}{x+1} \rightarrow u = 2\]

Answer 2

Would this be the transformation?