For a constant define

Find the values for such that the region above the graph of and below the graph of has area equal to .

We consider three cases: , and .

Case 1: . This is not possible since if then and so the area above the graph of and below the graph of is equal to

Case 2: . If then on , so we have the area, , of the region between the two graphs given by

Setting this equal to and solving for we have

Case 3: . If then on so

Setting this equal to and solving for we obtain

Thus, the possible values of to make the area of the region above the graph of and below the graph of are .

Hi! First of all, thank you so much for your help with these solutions! I have a quick question about this exercise. The statement says ‘the region above the graph of g and below the graph of f.’ Then, shouldn’t we assume the case f > g?

Hi! Quick question, how did you know that the interval was when ?

Hi! Since we want to look at the area below and above we want the interval to correspond to where . So, we look for

We have equality if . If then this gives us . Since we have in this case by assumption, we know so the interval is . So we have when .