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 $y+3$

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