Answer
$\dfrac{13}{24}$
Work Step by Step
In order to find the volume, we have:
$V=\int_{0}^{ \pi/2 }\int_{0} ^{1}\int_{0} ^{2-r \sin \theta } r^2 \cos \theta dx dr d\theta \\=\int_{0}^{ \pi/2}\int_{0} ^{1} 2r^2 \cos \theta -r^3 \cos \theta \sin \theta dr d\theta \\=\int_{0}^{ \pi/2}\int_{0} ^{1} \dfrac{2 \cos \theta}{3} - \dfrac{\cos \theta \sin \theta d\theta}{4} $
Plug $ a= \sin \theta $ and $du =\cos \theta d\theta$
Now, we have:
$V =\dfrac{2}{3} \times |\sin \theta|_0^{\pi/2} -\int_0^1 \dfrac{a }{4} da =\dfrac{13}{24}$
