Find the area, lying above x - axis and included between the circle circle x^{2} + y^{2} = 8x and the parabola y^{2} = 4x.

To find area lying above x - axis and included in the circle

x^{2} + y^{2} = 8x ...(i)

Or (x – 4)^{2} + y^{2} = 16

And ...(ii)

On solving the equation (i) and (ii),

x^{2} + y^{2} = 8x

Or x^{2} – 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 ODBO

Region OBDO = 32/3 sq. units …(2)

Region DABD

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 x^{2} + y^{2} = 8x and the parabola y^{2} = 4x is 4 sq. Units

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