The area of the region, inside the ellipse x^2 + 4y^2 = 4 and outside the region bounded b…

Question

The area of the region, inside the ellipse $x^2 + 4y^2 = 4$ and outside the region bounded by the curves $y = |x| - 1$ and $y = 1 - |x|$, is:

Accepted Answer

The ellipse $E : \dfrac{x^2}{4} + \dfrac{y^2}{1} = 1$ has semi-axes $2$ and $1$, so its area is $\pi(2)(1) = 2\pi$.

The lines $y = |x|-1$ and $y=1-|x|$ bound a square (diamond) $PQRS$ with vertices at $(\pm1,0)$, $(0,\pm1)$, whose diagonal is $2$, so its side is $\sqrt2$ and its area is $(\sqrt2)^2 = 2$.

Required area $= 2\pi - 2 = 2(\pi - 1)$.

Solutions · 1