Question

A circle of radius 1 is tangent to the parabola y=x^2 as shown. Find the gray area between the circle and the parabola?

Answer 1

Equation of the circle = x^2 + (y - k)^2 = 1 ----- c {k is the 'y' coordinate of the centre of the circle}
Equation of the parabola y = x^2 ----- p
at the point of contact, the slopes of both the equations must be the same.

y=x^2
(dy/dx) = 2x {Parabola} ---- 1

y = -(1 - x^2)^0.5 + k (we just need the lower part of the circle to calculate the area}
(dy/dx) = -(1/(2(1 - x^2)^0.5))*(-2x) = x/(1 - x^2)^0.5 {Circle} ------ 2

1 = 2

2x = x/(1 - x^2)^0.5
(1 - x^2)^0.5 = 1/2
1 - x^2 = 1/4
x^2 = 3/4
x = 3^0.5/2 ---- 3

Sub 3 in p
y = x^2 = 3/4 = 0.75 ---- 4

Sub 4 and 3 in y = -(1 - x^2)^0.5 + k
0.75 + (1 - 3/4)^0.5 = k
k = 1.25

Therefore the coordinates of the intersection are (3^0.5/2, 0.75) & (-3^0.5/2, 0.75).

Required area = integration of ( -(1 - x^2)^0.5 + 1.25 - x^2) from x = -3^0.5/2 to x = 3^0.5/2. = 0.251841 Sq units.