Area between function w/ definite integrals

[asset8789]

Would somebody mind explaining this problem to me? I am fairly certain that III would do the trick but I have no idea how or what to do to determine if I or II are functional answers to this.

 

[Zermelo]

You should first ask yourself, what is a definite integral in terms of area? What would be the integral of the function y=3? What would be the integral of the function f?
They are both areas UNDER the curve. Now, the question is, how to compute the shaded area, while knowing the areas under these curves? Try to draw the problem out, and solve it on your own, then, by using some properties of integrals, one of the other answers should correspond to your solution.

 

[pka]

Suppose that [imath]\Phi[/imath] is an integrable function on [imath][a,b][/imath].
Do you understand that [imath]\displaystyle\int_a^b {\Phi (x)dx} = \int_b^a {-\Phi (x)dx} [/imath]