The respective equations for the parabola and the circle are:
y2 = 4x ... (1)
4x2+4y2 = 9 ... (2)
or x2+y2 = (

3

2

)2
Equation (1) is a parabola with vertex (0, 0) which opens to the right and equation (2) is a circle with centre (0, 0) and radius

3

2

From equations (1) and (2), we get:
4x2 + 4 (4x) = 9
4x2 + 16x - 9 = 0
4x2 + 18x - 2x - 9 = 0
2x (2x + 9) - 1 (2x + 9) = 0
(2x + 9) (2x - 1) = 0
x = -

9

2

,

1

2

For x = −

9

2

, y2 = 4 (−

9

2

) , which is not possible, hence x =

1

2

Therefore, the given curves intersect at x =

1

2

Required area of the region bound by the two curves
= 2

1 2

∫

0

2√xdx + 2‌

3 2

∫

1 2

√

9

4

−x2dx
= 4 |

2

3

x

3

2

|0

1

2

+ 2
=

8

3

(

1

8

)

1

2

+ 2
=

8

3

(

1

2√2

) +

9

4

(

π

2

) -

√2

2

−

9

4

sin−1(

1

3

)
=

2√2

3

+

9π

8

-

√2

2

−

9

4

sin−1(

1

3

)