You can begin to solve this problem by setting the two equations equal to each other in order to figure out the points in which they intersect. You will find these points to be (0, 2) and (5, 2).
Then, plug in any x-value in that range (from 0 to 5) into both equations to see which equation yields the largest value. This equation, which will be y = -x^2 + 5x + 2, is always above the other equation, y = 2, on the interval from 0 to 5 noninclusive.
Now you can set up an integral that goes from 0 to 5. The expression inside the integral will be the top equation, which we found to be y = -x^2 + 5x + 2, minus the bottom equation, y = 2, meaning that the resulting integral will be:
∫ (-x^2 + 5x) dx, going from 0 to 5.
Once you integrate that, you will get
-(1/3)x^3 + (5/2)x^2 going from 0 to 5, and the resulting answer is 20.833.