`x=2y^2, x=4+y^2`

`2y^2=4+y^2`

`2y^2-y^2=4`

`y^2=4`

`y=+-2`

Refer the attached image. Graph of x=2y^2 is plotted in red color and graph of x=4+y^2 is plotted in blue color.

Area of the region enclosed by the curves A=`int_(-2)^2((4+y^2)-2y^2)dy`

`A=2int_0^2(4-y^2)dy`

`A=2[4y-y^3/3]_0^2`

`A=2(4*2-2^3/3)`

`A=2(8-8/3)`

`A=2((24-8)/3)`

`A=32/3`

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`x=2y^2, x=4+y^2`

`2y^2=4+y^2`

`2y^2-y^2=4`

`y^2=4`

`y=+-2`

Refer the attached image. Graph of x=2y^2 is plotted in red color and graph of x=4+y^2 is plotted in blue color.

Area of the region enclosed by the curves A=`int_(-2)^2((4+y^2)-2y^2)dy`

`A=2int_0^2(4-y^2)dy`

`A=2[4y-y^3/3]_0^2`

`A=2(4*2-2^3/3)`

`A=2(8-8/3)`

`A=2((24-8)/3)`

`A=32/3`