Find the area of the region common to the circle x2 + y2 = 16 and the parabola y2 = 6x.
There are two equations,
x2 + y2 = 16 ...(i)
y2 = 6x ...(ii)
From (i) and (ii)
X2 + 6x = 16
Or, X2 + 6x – 16 = 0
Or, (x + 8)(x – 2) = 0
Or, x = – 8 or x = 2
And when x = 2 then y = ±2√3
Equation (i) represents a circle with centre (0,0) and radius 4 units, so it meets x - axis at (±4,0) and equation (ii) represents a parabola with vertex (0,) and axis as x - axis
Points of intersection of parabola and circle are (2,2√3) and (2, – 2√3).
The sketch of the two curves are drawn below,
The shaded region represents the required area.
Required area = Region OBCAO
Required area = 2 (region ODAO + region DCAD)
So, the required area is sq units.