Question

Find the area of the largest rectangle that will fit inside the region bounded by y = - x^2 +1 and y = x^2 ?

Answer 1

If A is the area of the required rectangle between the 2 parabola, then
A/2 = (-x^2 + 1 - x^2)×x
A/2 = -2x^3 + x ---- 1
Now differentiating wrt x and equating it to 0
-6x^2 + 1 = 0
x = +/-(1/(6^0.5)) --- 2
Sub 2 in 1
A/2 = 2/(3×(6^0 5))
A = 4/(3×(6^0.5)) is the max area possible for a rectangle between the 2 parabolas with dimensions (2/6^0.5) & (2/3).