Question

let i=∫∫d(x²−y²)dxdy , where d={(x,y): 3≤xy≤4, 0≤x−y≤5,x≥0, y≥0}. show that the mapping u=xy , v=x−y maps d to the rectangle r=[3,4]×[0,5] .

Answer 1

To show that the mapping u = xy, v = x - y maps d to the rectangle r = [3,4] x [0,5], we need to demonstrate that the region of integration over d corresponds to the region in r.

Step-by-step explanation:

To show that the mapping u = xy, v = x - y maps d to the rectangle r = [3,4] x [0,5], we need to demonstrate that the region of integration over d corresponds to the region in r.

First, we need to find the limits of integration for x and y in terms of u and v.

By solving the equations u = xy and v = x - y for x and y, we get x = (u + v) / 2 and y = (u - v) / 2.

Substituting these expressions into the conditions for d, we have 3 ≤ (u² - v²) / 4 ≤ 4 and 0 ≤ (u + v) / 2 - (u - v) / 2 ≤ 5. Simplifying these inequalities, we obtain 3u - v ≤ 8 and -u + 3v ≤ 10.

These inequalities define the region in the uv-plane that corresponds to the region d in the xy-plane, which is the rectangle r = [3,4] x [0,5].