Given a function , gives us the area between the function and the x-axis, from a to b.
Naturally, when we have two functions and , where is always bigger than , then gives us the area between the two functions.
So, we have 3 points . We can find three lines from these points, and each line will cross two of the points. These will be the sides of the triangle. We can obtain these with simple algebra
crosses (0,0) and (2,1)
crosses (0,0) and (-1,6)
crosses (2,1) and (-1,6)
Now if we draw out the triangle on the xy plane, we see that is above when
So the area between the two lines is given by .
Similarly, is about when and the area between the two lines is given by .
I'll leave you to solve these integrals. Now we see that by our setup, the sum of the two areas, is the area of the triangle. So the area of the triangle is