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SithsNGiggles
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Homework Statement
Consider the area bounded between the curves [itex]y=3-x^2[/itex] and [itex]y=-2x[/itex]. Suppose two vertical lines, one unit apart, intersect the given area. Where should these lines be placed so that they contain a maximum amount of the given area between them? What is this maximum area?
Homework Equations
The Attempt at a Solution
First I computed the area between the two curves using the integral
[itex]A = \int_{-1}^{3} (3 - x^2) - (-2x) dx[/itex]
[itex]A = \int_{-1}^{3} 3 + 2x - x^2 dx[/itex]
[itex]A = \frac{32}{3}[/itex].
Then I set up another integral for the maximum area between two vertical lines [itex]y = k[/itex] and [itex]y = k+1[/itex]:
[itex]A(k) = \int_{k}^{k+1} 3 + 2x - x^2 dx[/itex], for [itex]k \in [-1, 3][/itex].
I'm not sure where to go from here, but something tells me there's some application of the first derivative test that I can put to use here. Unfortunately I'm not sure how to apply it. Any help is appreciated, thanks.