Aregion r in the xy-plane is given. find equations for a transformation t that maps a rectangular region s in the uv-plane onto r, where the sides of s are parallel to the u- and v- axes. (let u play the role of r and v the role of θ. enter your answers as a comma-separated list of equations.) r lies between the circles x2 + y2 = 1 and x2 + y2 = 7 in the first quadrant

X(u, v) = (2(v - c) / (d - c) + 1)cos(pi * (u - a) / (2b - 2a))
y(u, v) = (2(v - c) / (d - c) + 1)sin(pi * (u - a) / (2b - 2a))

As v ranges from c to d, 2(v - c) / (d - c) + 1 will range from 1 to 3, which is the perfect range for the radius. As u ranges from a to b, pi * (u - a) / (2b - 2a) will range from 0 to pi/2, which is the perfect range for the angle. So, this maps the rectangle to R.