 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/12/b/i/solution

The region lies above the parabola

y = x^{2}

and below the upper semicircle

y = 1 - x^{2}

. Their intersections satisfy

x^{4} + x^{2} - 1 = 0

, hence

x = \pm a

with

a^{2} = (5 - 1) /2

. In particular,

1 - a^{2} = a^{2}

. The region is

D = {(x, y) : - a \leq x \leq a, x^{2} \leq y \leq 1 - x^{2}} .

It is symmetric about the

y

axis, with endpoints

(\pm a, a^{2})

; at

x = 0

it extends from

y = 0

y = 1

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