Question

If the area of the region $\\{(x, y) : x^{2/3} + y^{2/3} \leq 1, x + y \geq 0, y \geq 0\\}$ is A, then $\frac{(256 A)}{\pi}$is equal to_______.

Answer 1

$∴$ Area of shaded region =$$\int_{-\frac{1}{2}^{3/2}}^0 \left((1 - x^{2/3})^{3/2} + x\right) \, dx + \int_0^1 (1 - x^{2/3})^{3/2} \, dx

=$$\int_{-\frac{1}{2}^{3/2}}^0 (1 - x^{2/3})^{3/2} \, dx + \int_{-\frac{1}{2}^{3/2}}^0 x \, dx
Let x $=$ $\sin^3 \theta$
$∴$ $dx = 3\sin^2 \theta \cos \theta \, d\theta$

$\int_{-\frac{\pi}{4}}^{\frac{\pi}{2}} 3\sin^2 \theta \cos^4 \theta \, d\theta + \left(0 - \frac{1}{16}\right)$

=$$\frac{9\pi}{64} + \frac{1}{16} - \frac{1}{16} = \frac{36\pi}{256} = A

∴$$\frac{256A}{\pi} = 36