Thread: Maximum area of an area within y=x^2

1. 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!

2. if you make the left hand bottom corner of the rectangle point a then it follows that

$A = L\times W$

subing in some values gives

$A = f(a)\times (2-a)$

$A = a^2\times (2-a)$

$A = 2a^2-a^3$

for a maximum you must find where $\frac{dA}{da}=0$

$\frac{dA}{da}=4a-3a^2$

$0=4a-3a^2$

$0=a(4-3a)$

$a=0,\frac{4}{3}$

probably should discard zero as a solution at this point as it implies zero area.

now $f\left(\frac{4}{3}\right)= \frac{9}{16}$

Maximum area should be simple to find from here using

$A = L\times W$

$A = f(a)\times (2-a)$