The parabolas `y^2=4xa n dx^2=4y` divide the square region bounded by the lines `x=4,y=4` and the coordinate axes. If `S_1,S_2,S_3` are the areas of these parts numbered from top to bottom, respectively, then `S_1: S_2-=1:1` (b) `S_2: S_3-=1:2` `S_1: S_3-=1:1` (d) `S_1:(S_1+S_2)=1:2`

Options

A. `S_(1):S_(2)equiv1:1`
B. `S_(2):S_(3)equiv1:2`
C. `S_(1):S_(3)equiv1:1`
D. `S_(1):(S_(1)+S_(2))=1:2`

Answer

A, C, D

Explanation

`y^(2)=4x and x^(2)=4y" meet at "O(0,0) and A(4,4).`

`"Now, "S_(3)=int_(0)^(4)(x^(2))/(4)dx=(1)/(4)[(x^(3))/(3)]_(0)^(4)=(1)/(12)[64-0]=(16)/(3).`
`S_(2)=int_(0)^(4)2sqrt(x)dx-S_(3)=2[(x^(3//2))/(3//2)]-(16)/(3)`
`=(4)/(3)[8-0]-(16)/(3)=(16)/(3).`
`"And "S_(1)=4xx4-(S_(2)+S_(3))=16-((16)/(3)+(16)/(3))=(16)/(3).`
`"Hence, "S_(1):S_(2):S_(3)=1:1:1`