I am a teacher and such status does not protect me from being wrong. Nor does it do so for your teacher (else, he/she has one terrific contract!).
Now, picture the graph of y=x on the interval [-4, 4]. As the function is symmetric about the origin, the integral from -4 to 0 is necessarily opposite that from 0 to 4. The sum of two opposites is, of course, zero.
it's me again. If the answer isn't zero the only possibility I can think of is that you should calculate the area which is enclosed by y = x and x = -2 and x = 2.
The straight line with the equation y = x has a zero at x = 0. Thus you have to split the integral into 2 parts: