If the area of the region (x,y):x^2/3+y^2/31, x+y0, y0 is A, then 256A is equal to_____ An…

Question

If the area of the region $\left\{(x,y):x^{2/3}+y^{2/3}\le1,\ x+y\ge0,\ y\ge0\right\}$ is $A$, then $\dfrac{256A}\pi$ is equal to\_\_\_\_\_

Answer: 36

Accepted Answer

Using the substitution $x=\sin^3\theta$ for the astroid arc together with the triangular piece cut off by $x+y\ge0$, the area works out to
$$A=\frac{9\pi}{64}+\frac1{16}-\frac1{16}=\frac{36\pi}{256}.$$

$$\frac{256A}\pi=36.$$

Solutions · 1