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Question
Find the area of the sector of a circle bounded by the circle x2 + y2 = 16 and the line y = x in the ftrst quadrant.
Solution
Given that x2 + y2 =16 ...(i)
y=x ...........(ii)
By equation (i) & (ii)
`x=+-2sqrt2`
`y=+-2sqrt2`
But required area in first quadrant
`x= y=2sqrt2`
`From dig. area = Area of Δ OBC + Area of region CABC`
`=int_o^(2sqrt2)x dx+int_(2sqrt2)^4sqrt(16-x^2)dx`
`=1/2 [x^2]_0^(2sqrt2)+[x/2sqrt(16-x^2)+16/2 sin^-1 (x/4)]_sqrt2^4`
`=4+8 xxpi/2-4-8xxpi/4=2pi sq.units`
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