Question

Find the volume of the solid generated by revolving the region enclosed by the triangle with vertices

(2,2), (2,4), and (6,4) about the y-axis.
a) 24 units³
b) 36 units³
c) 48 units³
d) 72 units³

Answer 1

To find the volume of the solid generated by revolving the region enclosed by the triangle about the y-axis, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region enclosed by the triangle about the y-axis, we can use the method of cylindrical shells.

1. First, we need to calculate the height of the cylinder, which is the distance between the two vertices of the triangle along the y-axis. In this case, it is 4 - 2 = 2 units.
2. Next, we need to calculate the radius of each cylindrical shell. This can be done by finding the distance between the y-axis and each point on the triangle. Since the triangle is vertical and the y-axis is the axis of revolution, the distance is simply the x-coordinate of each point. For the points (2,2) and (2,4), the distances are 2 and 2 respectively. For the point (6,4), the distance is 6.
3. Therefore, the radius of each cylindrical shell is given by:

• r = 2 (for the points (2,2) and (2,4))
• r = 6 (for the point (6,4))

Finally, we can calculate the volume of each cylindrical shell using the formula V = 2πrh, where r is the radius and h is the height. Then, we can add up the volumes of all the cylindrical shells:

V = (2π(2)(2)) + (2π(2)(2)) + (2π(6)(2)) = 24π + 24π + 24π = 72π

So the volume of the solid generated by revolving the region enclosed by the triangle about the y-axis is 72π. Since the question asks for the answer in units³, we can approximate π as 3.14, giving us a volume of approximately 226.08 units³.