Find the area bounded by the curves `x^2+y^2=25 ,4y=|4-x^2|,` and `x=0` above the x-axis.

Options

A. `4+25 sin^(-1) (4//5)`
B. `(1)/(2){8-25sin^(-1)(4//5)}`
C. `(1)/(4) { 4 25 sin^(-1)(4//5)}`
D. None of these

Answer

C

Explanation

`4y=|4-x^(2)|.xge0`
`rArr {{:(4y=4-x^(2),:0 lex le2),(4y=x^(2)-4,:xge2):}`

Region bounded by the curves
`x^(2)+y^(2)=25`
`4y=|4-x^(2)|,xge0,ygt0` is shaded
Solving `x^(2)=4(y+1) and x^(2)+y^(2)=25` we, get
`y^(2)+4y-21=0`
`rArr y=3`
`rArr x=4`
Required area
`int _(0)^(2)(sqrt(25-x^(2))-(1)/(4)(4-x^(2)))dx`
`+int _(2)^(4)(sqrt(25-x^(2))-(1)/(4)(x^(2)-4))dx`
`=(1)/(2)(4+25+ sin^(-1)(4//5))`