The reason why we can do such operations for this integral is that int (1 0 f(x) dx) represents the area under the graph of f(x) from x = 0 to x = 1 and int (4 1 f(x) dx) represents the area under the graph of f(x) from x = 1 to x = 4. Of course, summing up the areas gives us a total area equivalent to the area under the graph of f(x) from x = 0 to x = 4, which is represented as int (4 0 f(x) dx).