First find where the line x+y=5 meets the hyperbola xy=4, so y=4/x.
Substitute y=5-x in xy=4, then x(5-x)=4, 5x-x²=4, so:
x²-5x+4=0=(x-4)(x-1). So the limits of the bound area are 1 and 4 (for x).
Let y₁=5-x and y₂=4/x.
The hyperbola sits below the line so the integral to find the area is:
∫⁴₁(y₁-y₂)dx=∫⁴₁(5-x-4/x)dx=
(5x-x²/2-4ln(x))|⁴₁=(20-8-4ln(4))-(5-1/2)=
7.5-4ln(4)=1.9548 approx.