Volume in the first octant

Geometry Level 3

What is the volume of the portion of the cylinder $x^{2}$ + $y^{2}$ = 4 in the first octant of solid geometry between the planes z=0 and 3x-z=0 ?

1 solution

Tom Engelsman
Jun 3, 2018

Let $0 \le \theta \le \frac{\pi}{2}; 0 \le r \le 2; 0 \le z \le 3r \cdot cos \theta$ . The required volume computes to:

$V = \int_{0}^{\frac{\pi}{2}} \int_{0}^{2} \int_{0}^{3r cos \theta} r dzdrd\theta = \boxed{8}.$