Find the volume of the solid obtained when the region bounded by the line $y = 2 x$ , the line $x = 3$ , and the x-axis is rotated about the y-axis.
A) $36 \pi$
B) $42 \pi$
C) $16 \pi$
D) $18 \pi$
E) $24 \pi$
F) $30 \pi$
G) $48 \pi$
H) $27 \pi$

Question

Find the volume of the solid obtained when the region bounded by the line $y = 2 x$ , the line $x = 3$ , and the x-axis is rotated about the y-axis.
A) $36 \pi$ 
B) $42 \pi$ 
C) $16 \pi$ 
D) $18 \pi$ 
E) $24 \pi$ 
F) $30 \pi$ 
G) $48 \pi$ 
H) $27 \pi$

Quizplus · Accepted Answer

The volume of the solid obtained by rotating the given region around the y-axis can be calculated using the disk method. The region is bounded by $y = 2x$, $x = 3$, and the x-axis. When rotated about the y-axis, the radius of a disk at a given y-value is $x = \frac{y}{2}$, and the height of the disk is an infinitesimally small change in y, $dy$. The volume of each disk is $\pi r^2 dy = \pi (\frac{y}{2})^2 dy$. The bounds for y are from 0 to 6 (since $y = 2(3)$). Integrating from 0 to 6 gives the total volume: $\int_{0}^{6} \pi (\frac{y}{2})^2 dy = \int_{0}^{6} \frac{\pi y^2}{4} dy = \frac{\pi}{4} \left[\frac{y^3}{3}\right]_0^6 = \frac{\pi}{4} \cdot \frac{6^3}{3} = 36\pi$.

Find the Volume of the Solid Obtained When the Region y=2xy = 2 xy=2x

Find the Volume of the Solid Obtained When the Region $y = 2 x$