Proving the area of a shaded rectangle under a parabola and then differentiating the expression.
Minimizing the perimeter of sectors in an arc.
Please see the attached file for the fully formatted problems.
because parabola eqn. is given as:
y = 3 - x^2
Rectangle length is 2x and is symmetric about y axis, it means, the intersection points of parabola and rectangle are:
(x, 3-x^2) and (-x, 3-x^2)
therefore the area of shaded part:
A = area of the rectangle = length * width
width = ...
The area of a rectangle under a parabola is calculated and the perimeter of a sector in an arc is minimized.