area between the curves y^2=4x , 7y=2x+20 , 2x+3y=0
I guess, looking at the graphs at
http://www.wolframalpha.com/input/?i=plot+y^2%3D4x,7y%3D2x%2B20,2x%2B3y%3D0
that you want the triangular-ish region with vertices at
(-3,2), (4,4), (0,0)
It looks like horizontal strips will be easiest to use, so the area is
∫[0,2] (y^2/4)-(-3y/2) dy
+ ∫[2,4] (y^2/4)-((7y-20)/2) dy
= 11/3 + 11/3
= 22/3