Question

Let R be the region in the first quadrant of the xy-plane bounded by the hyperbolas xy=1,xy=16, and the lines y=x,y=25x. Use the transformation x=u/v ,y=uv with u>0 and v>0 to rewrite the integral below over an appropriate region G in the uv-plane. Then evaluate the uv-integral over G.

∬ᴿ (√y/x +√xy)dxdy

Answer 1

The rewritten integral over the region G in the uv-plane is ∬ᴳ (u+v) dudv = 3/2 ln(4).

Step-by-step explanation:

To rewrite the given integral over the region G in the uv-plane, we first need to transform the variables x and y into u and v using the given transformation x=u/v and y=uv. The region R in the xy-plane bounded by the hyperbolas xy=1 and xy=16, and the lines y=x and y=25x, transforms into a region G in the uv-plane. After performing the transformation, the integral becomes ∬ᴳ (u+v) dudv. Evaluating this integral over G yields 3/2 ln(4).

The transformation x=u/v and y=uv maps the region R onto a new region G in the uv-plane. The boundaries of R in terms of u and v are found by solving for u and v in terms of x and y. After obtaining the boundaries for G, we can rewrite the given integral in terms of u and v. Evaluating this new integral over G gives us the final answer of 3/2 ln(4).

The evaluation of the uv-integral over G involves integrating (u+v) with respect to u and v over their respective boundaries. After performing these integrations, we arrive at the final result of 3/2 ln(4).