Find the area of the region using the method that requires you to evaluate just one integral.

Region between y=x and x+y=8 over [2,3]

First graph the two functions to make sure that the required region is positive within the limits of integration.

Let
y1(x)=x
y2(x)=8-x
y1(2)=2, y1(3)=3
y2(2)=6, y2(3)=5.
Thus the region is always positive (the two function do not cross within the interval [2,3]).

The incremental area is
(y2(x)-y1(x))dx
The area is thus the integral
∫(y2(x)-y1(x))dx between the limits 2 and 3.