Find the area, lying above x - axis and included between the circle circle x2 + y2 = 8x and the parabola y2 = 4x.
To find area lying above x - axis and included in the circle
x2 + y2 = 8x ...(i)
Or (x – 4)2 + y2 = 16
On solving the equation (i) and (ii),
x2 + y2 = 8x
Or x2 – 4x = 0
Or x(x – 4) = 0
Or x = 0 and x = 4
When x = 0, y = 0
When x = 4, y = ±4
Equation (i) represents a circle with centre (4,0) and meets axes at (0,0) and (8,0).
Equation (ii) represent a parabola with vertex (0,0) and axis as x - axis.
They intersect at (4, – 4) and (4,4).
These are shown in the graph below: -
Required area = Region OABO
Required area = Region ODBO + Region DABD …(1)
Region OBDO = 32/3 sq. units …(2)
Region DABD = 4π sq. units …(3)
Using (1),(2) and (3), We get
Required area =
= 4 sq. units
The area lying above the x - axis and included between the circle x2 + y2 = 8x and the parabola y2 = 4x is 4 sq. Units