Find the area of the region {(x, y) : y^{2} ≤ 4x, 4x^{2} + 4y^{2} ≤ 9}

The area bounded by the curves, {(x, y) : y^{2} ≤ 4x, 4x^{2} + 4y^{2} ≤ 9}, is shown by shaded region as OABCO.

The points of intersection of both the curves are

We can observed that area OABCO is symmetrical about x-axis.

Thus, Area of OABCO = 2 × Area OBC

Now,

Area OBCO = Area OMC + Area MBC

⇒

⇒

Put 2x = t

⇒ dx =

So, when x = , t = 3 and x = , t = 1, we get,

⇒

⇒

⇒

⇒

⇒

⇒

⇒

Therefore, the required area is

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