You did that much correctly.
Now, you want to find the INTERSECTION point of the two curves. That is, you want to know where the curves have the same y value? Is that not where g(x) == h(x)?
Using fzero on g alone will not help you! What is the root that fzero can find that will help? How about this? If we want to find the point where g==h, then what if we solve for the point where (g(x)-h(x))==0?
fzero(@(x) g(x) - h(x),-0.2)
Once you have done this for BOTH intersections, now what area do you want to find? You want to find the area of g(x) - h(x), since in that interval, g(x) is greater than h(x).