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Find the area bounded by the curves x^2+...

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

Text Solution

Verified by Experts

The correct Answer is:
`[4+25 sin^(-1)((4)/(5))]` sq units
Given curves , `x^(2) +y^(2)=25,4y=|4-x^(2)|` could be sketched as below, whose points of intersection are
`x^(2)+((4-x^(2))^(2))/(16)=25`

`rArr (x^(2)+24)(x^(2)-16)=0`
`rArr x = pm 4`
`therefore " Required area "=2[int_(0)^(4)sqrt(25-x^(2))dx -int_(0)^(2)((4-x^(2))/(4))dx - int_(2)^(4) ((x^(2)-4)/(4))dx]`
`=2[[(x)/(2)sqrt(25-x^(2))+(25)/(2) sin^(-1)((x)/(5))]_(0)^(4)-(1)/(4)[4x-(x^(3))/(3)]_(0)^(2)-(1)/(4)[(x^(3))/(3)-4x]_(2)^(4)]`
`=2[[6+(25)/(2) sin^(-1)((4)/(5))] (1)/(4)[8-(8)/(3)]-(1)/(4)[((64)/(3)-16)-((8)/(3)-8)]]`
`=2[6+(25)/(2) sin^(-1)((4)/(5))-(4)/(3)-(4)/(3)-(4)/(3)]`
`=[4+25 sin^(-1)((4)/(5))]` sq units
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