Draw a rough sketch of the region {(x, y) : y² ≤ 6ax and x² + y² ≤ 16a²}. Also find the area of the region sketched using method of integration.

Given;

The region {(x, y) : y² ≤ 6ax and x² + y² ≤ 16a²}

By solving the equations: y² ≤ 6ax and x² + y² ≤ 16a²

Through substituting for y²

⇒ x² + 6ax = 16a²

⇒ (x − 2a) (x + 8a) = 0

∴ x = 2a.

[as x = -8a is not possible]

Area of the region bounded by the curve y=f(x), the x-axis and the ordinates x=a and x=b, where f(x) is a continuous function defined on [a,b], is given by .

[By the symmetry of the image w.r.t x axis]

Required area =