Find the area of the region {(x, y) : y2 ≤ 4x, 4x2 + 4y2 ≤ 9}

Question

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

Philoid · Accepted Answer

The area bounded by the curves, {(x, y) : y² ≤ 4x, 4x² + 4y² ≤ 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

Find the area of the region {(x, y) : y² ≤ 4x, 4x² + 4y² ≤ 9}