Well, you may be able to convince your teacher with an explicit solution, and yourself that such a situation is perfectly normal:
You have two values for an integral of an unknown function f(x), namely from 1 to 4 and from 1 to 6. If you give f(x) two parameters (call them a and b) and can find the integral for the two intervals given, then you can determine f(x) and hence do the integral from 4 to 6. The trick is to find a simple enough function f(x) so this can actually be done. The simplest case I can come up with is a straight line: f(x)=a*x+b.
Then the calculation goes like this:
f:=(x) -> a*x+b;
x -> a x + b
1 / 2 \
- a \X - 1/ + b (X - 1)
[[ -23 37]]
[[a = ---, b = --]]
[[ 15 6 ]]
- -- x + --
confirming your assertion. The plot looks like this:
I hasten to add that f(x) is not uniquely defined by the integral values; there will be infinitely many different ones with the same integrals. But since the integrals are additive (rule # 4 in your book), the result is in fact general and not dependent on the details of f(x).
I hope this helps,