Q. 204.1( 7 Votes )

Find the area, lying above x - axis and included between the circle circle x2 + y2 = 8x and the parabola y2 = 4x.

Answer :

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

x2 + y2 = 8x ...(i)


Or (x – 4)2 + y2 = 16


And ...(ii)


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 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 x2 + y2 = 8x and the parabola y2 = 4x is 4 sq. Units


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