What is the area between two lines?

Question

Find the area between the line #y=x-3# and the parabola #x=9-y^2#

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Answer 1 · 2018-05-24T21:04:11

Reflect your thinking to integrate horizontally.

The points of intersection are #(0,-3)# and #(5,2)#

graph{(x-y-3)(9-x-y^2)=0 [-8.535, 13.965, -6.03, 5.22]}

Integrate from the lesser #y# to the greater (from #-3# to #2#)

the greater #x# (the one on the right is #9-y^2# minus the lesser #x# (the one on the left is #y+3#

#int_-3^2 ((9-y^2)-(y+3)) \ dy = int_-3^2 (6-y-y^2) \ dy = 125/6#