Volume of My Valentine

Using the graph above the volume using the shell method is $V = 2\pi * \dfrac{4}{5} \int_{0}^{1} x((\sqrt{x} + \sqrt{1 - x^2}) - (\sqrt{x} - \sqrt{1 - x^2})) dx = \dfrac{16\pi}{5} \int_{0}^{1} x\sqrt{1 - x^2} dx$

Letting $x = \sin(\theta) \implies dx = \cos(\theta) d\theta \implies V = -\dfrac{16\pi}{5} \int_{0}^{\frac{\pi}{2}} (\cos(\theta))^2 (-\sin(\theta)) d\theta = \dfrac{-16\pi}{15} \cos^3(\theta)|_{0}^{\frac{\pi}{2}} = \dfrac{16\pi}{15} \approx \boxed{3.351032}$