f(x)=x^2 - 5x +7 and g(x) 3/x

Question

1) Indicate points of intersection
2) compute the area enclosed by the graphs of f(x) and g(x).

1) (1, 3) , (3, 1)

2)Area = ∫ [1 to 3] [(3/x) - (x^2 - 5x + 7)] dx
= [3 ln x - (1/3)x^3 + (5/2)x^2 - 7x
= 3 ln 3 - 8/3

am i doing the right thing here?

Looks to me like you've got it.