Find the volume of the solid generated by revolving the region about the y-axis.
-The region enclosed by the triangle with vertices $( 0,0 ) , ( 1,0 ) , ( 1,2 )$
A) $\frac { 4 } { 3 } \pi$
B) $4 \pi$
C) $\frac { 2 } { 3 } \pi$
D) $\frac { 1 } { 3 } \pi$

Question

Quizplus · Accepted Answer

The volume of the solid generated by revolving the triangle around the y-axis can be found using the formula for the volume of a cone, $V = \frac{1}{3}\pi r^2 h$, where $r = 1$ and $h = 2$. Substituting these values gives $V = \frac{1}{3}\pi (1)^2 (2) = \frac{2}{3}\pi$. However, since the triangle forms a cone with a circular base that is revolved around the y-axis, the actual volume is twice this amount because the triangle's base spans from $y=0$ to $y=2$, effectively doubling the volume calculated for a single cone segment. Thus, the correct volume is $2 \times \frac{2}{3}\pi = \frac{4}{3}\pi$.

Find the Volume of the Solid Generated by Revolving the Region