Question

Solve the following:

Find the area enclosed between the circle x² + y² = 1 and the line x + y = 1, lying in the first quadrant.

Answer 1

Required area = area of the region ACBDA

= (area of the region OACBO) – (area of the region OADBO)   ...(1)

Now, area of the region OACBO

= area under the circle x² + y² = 1 between x = 0 and x = 1

= `int_0^1 y  dx`, where y² = 1 – x²,

i.e. y = `sqrt(1 - x^2)`, as y > 0

= `int_0^1 sqrt(1 - x^2).  dx`

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

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

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

= `π/(4)`   ...(2)

Area of the region OADBO = area under the line x + y = 1 between x = 0 and x = 1

= `int_0^1y  dx`, where y = 1 – x

= `int_0^1 (1 - x).  dx`

= `[x - x^2/2]_0^1`

= `1 - (1)/(2) - 0`

= `(1)/(2)`   ...(3)

Put the value of equations (2) and (3) in equation (1)

∴ Required area = `(pi/4 - 1/2)`sq units.