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July 17th, 2009, 02:30 PM   #1
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Maximum area of an area within y=x^2

I've got this problem that I'm trying to solve, I've been twisting my brain for several hours now.


An area 'A' is limited by the curve y=x^2, the x-axel and the line x=2

In this area is a rectangle with the area R, as seen in the figure.

Calculate the maximum relation between R and A in exact form.


If someone could take a look at this and help me, I would appreciate it very very much!
(sorry for my bad english)
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July 17th, 2009, 03:48 PM   #2
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Re: Maximum area of an area within y=x^2

Quote:
Originally Posted by media12
I've got this problem that I'm trying to solve, I've been twisting my brain for several hours now.


An area 'A' is limited by the curve y=x^2, the x-axel and the line x=2

In this area is a rectangle with the area R, as seen in the figure.

Calculate the maximum relation between R and A in exact form.


If someone could take a look at this and help me, I would appreciate it very very much!
(sorry for my bad english)
The height of the rectangle is y, the width is 2-x. Since y=x^2, the area is 2x^2 - x^3. Take derivatives and get extrema at 4x - 3x^2. The solutions are x=0 (min) and x=4/3 (max). At max, y=16/9, so area = 32/27.
The area under the parabola is 8/3 (I'll let you derive it).
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