Question

Find the area bounded by the parabola 8+2y-y^2 and the lines y=-1 y=3

Answer 1

This is most easily done by integrating on y, using horizontal strips.

a = ∫[-1,3] x dy
= ∫[-1,3] (8+2y-y^2) dy

Answer 2

I just noticed that you did not specify that the y-axis formed part of the boundary. Without that condition, the horizontal strips extend leftward forever.

Of course, you may have mixed up x and y in places, so ...