what is the height of a rectangle with its base on the x-axis is to be inscribed under the graph of y=2-x^2?

Question

largest possible area

Answer 1

the curve y=2-x^2 is even, so the x-axis will intercept the rectangle length in its middle point.
Let (a) be the x coordinate of the right side of the rectangel
Let (b) be the y coordinate of the top side of the rectangel

area = 2ab (as the length of the rectangle is 2a)
b=2-a^2
area=4a-2a^3
d area/da=4-6a^2
d area/da=0 so 4-6a^2=0
3a^2=2
a=(+/-)sqrt(2/3)
b=2-2/3=4/3
height=1.33

Answer 2

note that the curve is symmetric about x=0
suppose the rectange base is (-z,0) - (z,0)
the area A is 2z* (2-z^2)

A' = 2(2-z^2) + 2z(-2z) = 4 - 6z^2
for max, 4 = 6z^2
so z = sqrt(4/6) = sqrt(2/3)
and height is 2-z^2 = 2 - (2/3) = 4/3