alex83, to find the graph, I would write so the graph crosses the y-axis at (0, 0) and (0, 1). Since the coefficient of is negative, it is a parabola opening to the left going through those points.
More important is the fact that y= -x is the same as x= -y so the two graphs intersect when x= -y= y- y^2 or, adding y to both sides, 2y- y^2= y(2-y)= 0 so y= 0 or y= 2. The two graph intersect at (0,0) and (-2, 2).
Yes, you could do this integrating in terms of x, one integral from -2 to 0, the other from 0 to 1/4.
But it might be simpler to do the integral in terms of y (perhaps that was what calum was thinking of in his second post). Drawing a horizontal line from from x= -y to x= y- y^2, we see that the height of a rectangle (in the Riemann sum) is (y- y^2)- (-y)= 2y- y^2 and that the "width" is dy. The area of any such rectangle is (2y- y^2)dy and so the area is their "sum": .
That should be very easy.
What I mean to say is if you have two functions f(x) and g(x), the area between the functions is equal to the area between the two inverse functions and