Question

Using integration, find the area of the region bounded by the parabola y² = 4x and the circle 4x² + 4y² = 9.

Answer 1

Given curves are

y² = 4x....(i)

4x² + 4y² = 9......(ii)

⇒ x² + y² = (\(\frac 32\))²

Obviously, curve (i) is right handed parabola having vertex at (0, 0) and axis along +ve direction of x-axis while curve (ii) is a circle having centre at (0, 0) and radius \(\frac 32\).

Shaded region is required bounded region, which is symmetrical about x-axis.

For coordinate of intersection points 'A' or 'B'.

Hence, area of required region = 2 [Area OADO + Area DACD]

[Note: Equation of circle in standard form is 

(x - α)² + (y - β)² = r², where (α, β) is centre and r is radius.]