Let R'=K[x,y] (K is a field) and R=K+xR'
Even though (x)<(x,y) are prime ideals of R', (x) intersect R = (x,y) intersect R. So R' is not integral over R. What is the integral closure of R in R'.
First I do not see clearly why (x) intersect R equals (x,y) intersect R. I think of R as a polynomial ring over K with every nonconstant term having an x. If you mod out by the x's you are left with K[x,y], if you mod out by (x,y) you get something else.
Second I am not sure what the integral closure could be. Perhaps it is K[x,y] itself?