Easiest way to do this is to use shell method (I find).
Where b and a are the y values where the two functions intersect, and R is the equation for the height of the shell. Basically f(y) - g(y).
If you use the shell method you'd need 2 integrals
and you'd have to invert y = 2^x to get x = ln(y)
and y =3 -x^2 to get x = (3-y)^1/2
Don't know where defleurer got his results
The easiesies way is to use washers
outer radius 2^x+1 and inner radius 4-x^2
Ah, sorry. Shouldn't have rushed that answer, I suppose.
But, with shell method, the idea is to integrate (surface area) is it not? And your height would be f(y) - g(y) (by which I did mean inverting the functions). Your radius is then the distance you are from the axis of rotation. Were that axis of rotation simply y=0, than the radius would just be y. In this case, our axis of rotation is y=-1 and thus our radius should be y + 1. Thus, i'm still thinking this should work