Answer
$\approx \dfrac{136 \pi}{99}$
Work Step by Step
The equation of a bumpy sphere can be written as:
$ \rho=1+\dfrac{1}{5} \sin 6 \theta \sin 5 \phi$
The spherical coordinates system can be expresses as:
$x=\rho \sin \phi \cos \theta $ and $ y=\rho \sin \phi \sin \theta ; z=\rho \cos \phi$ and $\rho=\sqrt {x^2+y^2+z^2}$
or, $\rho^2=x^2+y^2+z^2$
Therefore,
$Volume= \iiint_{V} dV=\int_0^{\pi} \int_0^{2 \pi} \int_{0}^{1+\dfrac{1}{5} \sin 6 \theta \sin 5 \phi} \rho^2 \sin \phi d\rho \ d\theta d \phi $
By using a calculator, we have:
$Volume=\int_0^{\pi} \int_0^{2 \pi} \int_{0}^{1+\dfrac{1}{5} \sin 6 \theta \sin 5 \phi} \rho^2 \sin \phi d\rho \ d\theta d \phi \approx \dfrac{136 \pi}{99}$
