"Cardioidic" Apple

Key is to revolve this cardioid about the y-axis, its volume will be:

$V = \iint 2 \pi x dx dy$

Only the right half of the cardioid can be revolved to obtain the full volume, so $- \pi/2 < \theta < \pi/2$

$\begin{array}{l} V = 2 \pi \int_{- \pi/2}^{\pi/2} \int_{0}^{1-\sin\theta} r \cos\theta r dr d\theta \\ = \frac{2 \pi}{3} \int_{- \pi/2}^{\pi/2} \cos\theta (1-\sin\theta)^3 d\theta \\ = -\frac{2 \pi}{3}\Bigg[\frac{1}{4} \big( 1 - \sin\theta \big)^4 \Bigg]_{- \pi/2}^{\pi/2} \\ = - \frac{\pi}{6} \Bigg[ (1-\sin(\pi/2))^4 - (1-\sin(- \pi/2))^4 \Bigg] \\ =-\frac{\pi}{6} \Bigg[ 0 - (1-(-1))^4 \Bigg] \\ = - \frac{\pi}{6} \Bigg[ 0 - 16 \Bigg] \\ = \frac{8 \pi}{3} \approx \boxed{8.38} \\ \end{array}$