Hi, not sure how to set this integral up for my Calculus II homework.
"Given a graph that contains a parabola and a line, the equation of the parabola is x=(y-3)²/4 and the equation of the line is y=6-x. Find the area of the shaded region".
Are we supposed to find the points of intersection between these two equations?
Well, sure. Otherwise, how do you know the limits of integration?
I'm sure you found that the curves intersect at (1,5) and (9,-3)
So, using horizontal strips of width dy, the area is
∫[-3,5] (6-y) - (y-3)^2/4 dy = 64/3
You could use vertical strips of width dx, but then you have to allow for the two branches of the parabola, and divide the region where the boundary changes:
∫[0,1] (3+2√x)-(3-2√x) dx + ∫[1,9] (6-x)-(3-2√x) dx