The area (in sq.

Question

The area (in sq. units) of the region $A = \{(x,y) : |x|+|y|\le1,\ 2y^2\ge|x|\}$ is:

Accepted Answer

By symmetry, work in the region $x\ge0$, $y\ge0$ and multiply appropriately. The boundary curves $y=1-x$ (from $|x|+|y|=1$) and $2y^2=x$ meet where $2y^2=1-y \Rightarrow 2y^2+y-1=0 \Rightarrow y=\dfrac12$.

Using symmetry (four congruent lobes by reflection):

$$\text{Area} = 4\int_0^{1/2}2y^2\,dy + \frac12\cdot\text{Area}(\triangle PAB) = \frac56$$

(triangle correction term as in the source's decomposition of the diamond minus the parabolic cut-outs) gives the final area $\dfrac56$.

Solutions · 1