Question

Find the area of the region in the first quadrant bounded by the circle x² + y² = 4 and the X-axis and the line x = y√3.

Answer 1

For finding the points of intersection of the circle and the line, we solve

x² + y² = 4 ………(1)

and x = y√3 ……..(2)

From (2), x² = 3y²

From (1), x² = 4 – y²

3y² = 4 – y²

4y² = 4

y² = 1

y = 1 in the first quadrant.

When y = 1, r = 1 × √3 = √3

∴ the circle and the line intersect at A(√3, 1) in the first quadrant

Required area = area of the region OCAEDO = area of the region OCADO + area of the region DAED

Now, area of the region OCADO = area under the line x = y√3, i.e. y = x/√3 between x = 0

and x = √3

Area of the region DAED