Find the area enclosed between the circle x^{2} + y^{2} = 1 and the line x + y = 1, lying in the first quadrant

#### Solution

Given equation of the circle is x^{2} + y^{2} = 1 ......(i)

and equation of the line is x + y = 1 ......(ii)

∴ y = 1 – x

From (i), we get

y^{2} = 1 – x^{2}

∴ y = `sqrt(1 - x^2)` ......(iii) ......[∵ In first quadrant, y > 0]

Substituting (ii) in (i), we get

x^{2} + (1 – x)^{2} = 1

∴ x^{2} + 1 – 2x + x^{2} = 1

∴ 2x^{2} – 2x = 0

∴ 2x(x – 1) = 0

∴ x = 0 or x = 1

When x = 0, y = 1 and when x = 1, y = 0

Required area = area of the region ACBDA

= area of the region OACBO – area of the region OADBO

= area under the circle x^{2} + y^{2} = 1 – area under the line x + y = 1

= `int_0^1 sqrt(1 - x^2) "d"x - int_0^1 (1 - x) "d"x` .....[From (iii) and (ii)]

= `[x/2 sqrt(1 - x^2) + 1/2 sin^-1 (x)]_0^1 - [x - x^2/2]_0^1`

= `[0 + 1/2 sin^-1 (1) - 0] - [1 - 1/2 - 0]`

= `1/2(pi/2) - 1/2`

= `(pi/2 - 1/2)` sq.units