Find the volume of the solid obtained when the region bounded by the x-axis, the y-axis, and the line $y + x = 3$ is rotated about the x-axis.
A) $3 \pi$
B) $2 \pi$
C) $8 \pi$
D) $10 \pi$
E) $9 \pi$
F) $12 \pi$
G) $4 \pi$
H) $6 \pi$

Question

Find the volume of the solid obtained when the region bounded by the x-axis, the y-axis, and the line $y + x = 3$ is rotated about the x-axis.
A) $3 \pi$ 
B) $2 \pi$ 
C) $8 \pi$ 
D) $10 \pi$ 
E) $9 \pi$ 
F) $12 \pi$ 
G) $4 \pi$ 
H) $6 \pi$

Quizplus · Accepted Answer

The region bounded by the x-axis, the y-axis, and the line $y + x = 3$ forms a right triangle with legs of length 3 (since when $x=0$, $y=3$ and when $y=0$, $x=3$). When this region is rotated about the x-axis, it forms a cone with height 3 and base radius 3. The volume of a cone is given by $\frac{1}{3}\pi r^2h$, where $r$ is the radius and $h$ is the height. Substituting $r=3$ and $h=3$ gives $\frac{1}{3}\pi (3)^2(3) = 9\pi$.

Find the Volume of the Solid Obtained When the Region y+x=3y + x = 3y+x=3

Find the Volume of the Solid Obtained When the Region $y + x = 3$