Find the area of the region bounded by the parabola `y^(2)=2px` and `x^(2)=2py`.

Options

A. `(4p^(2))/3" sq units"`
B. `(5p^(2))/3" sq units"`
C. `(7p^(2))/3" sq units"`
D. `(8p^(2))/3" sq units"`

Answer

A

Explanation

We have, `y^(2)=2px" and " x^(2)=2py`
`:. Y=sqrt(2px)`
`rArrx^(2)=2p.sqrt(2px)`
`rArrx^(4)=4p^(2).(2px)`
`rArrx^(4)=8p^(3)x`
`rArrx^(4)-8p^(3)x=0`
`rArrx^(3)(x-8p^(3))=0`

`:." Required area "=int_(0)^(2p)sqrt(2px)dx-int_(0)^(2p)(x^(2))/(2p)dx`
`=sqrt(2p)int_(0)^(2p)x^(1//2)dx-1/(2p)dx`
`=sqrt(2p)[(2(x)^(3//2))/3]_(0)^(2p)-1/(2p)[(x^(3))/3]_(0)^(2p)`
`=sqrt(2p)[2/3.(2p)^(3//2)-0]-1/(2p)[1/3(2p)^(3)-0]`
`=sqrtsp(2/3 .2p^(3//2))-1/(2p)(1/38p^(3))`
`=sqrt(2p)((4sqrt2)/3p^(3//2))-1/(2p)(8/3p^(3))`
`=(4sqrt2)/3 .sqrt2p^(2)-8/6p^(2)`
`((16-8)p^(2))/6=(8p^(2))/6`
`=(4p^(2))/3" sq units"`