Consider the region bounded by the parabola y = 27 - x^2 and the x-axis. A rectangle of maximum possible area is inscribed in this region such that one of its sides lies on the x-axis. The area of the bounded region that lies outside this rectangle is

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The parabola y = 27 - x^2 intersects the x-axis at x = -3 3 and x = 3 3 . The total area of the region bounded by the parabola and the x-axis is: Total Area = _ -3 3 ^ 3 3 (27 - x^2) dx = 2 ₀^ 3 3 (27 - x^2) dx = 2 [ 27x - x^3 3 ]₀^ 3 3 = 2(81 3 - 27 3 ) = 108 3 Let the inscribed rectangle have vertices at (x, 0) , (-x, 0) , (x, 27-x^2) , and (-x, 27-x^2) for x > 0 . The area of this rectangle is A(x) = 2x(27 - x^2) = 54x - 2x^3 . To maximize the area, we set A'(x) = 0 : A'(x) = 54 - 6x^2 = 0 x^2 = 9 x = 3 The maxi