Indeed the natural logarithm of A is defined as the area under the curve f(x) = 1/x from x=1 to x equals the value A:
Note that this is defined only for A>0.
To find the area under the curve 1/x^2 you simply integrate:
So the area under the curve 1/x^2 from x = 1 to x = A is:
For the area under 2/x, it's simply two times the natural logarithm.
You posted this question in a pre-calculus forum, so if this look like hieroglyphics to you, don't worry - you'll get to in school soon enough!