Some flaws (as noted) but nice work and the right approach.
int(81/x^4, 1, infinity) = int(81/x^4, 1, N) + int(81/x^4, N, infinity). I will focus on the second part (the tail).
int(81/x^4, N, infinity) <= .01/Pi
Take the lim as b --> infinity from N to b, and we get,
-27/x^3 evaluated from N to b is 0 - (-27/n^3) = 27/n^3 <= .01/Pi
Solving for N I get: 20.3941...so I will use 21 for N. Therefore, if everything is right, I get this as a conclusion:
The volume is finite, and int(g(x)^2, x, 1, 21) will approximate the volume to within .01.
Anyone see any flaws?