Final answer:
To find the volume of the solid generated by revolving a triangular region, we can use the method of cylindrical shells. For each option (x-axis, y-axis, line x=2, line y=4), we need to determine the radius, height, and volume of each shell and then integrate to get the total volume.
Step-by-step explanation:
To find the volume of the solid generated by revolving the triangular region bounded by the curve y = 4/x³ and the lines x=1 and y=1/2, we need to use the method of cylindrical shells. Let's solve for each option:
a) Revolving about the x-axis:
- The radius of each shell is the distance from the x-axis to the curve, which can be expressed as r = y. Therefore, r = 1/2.
- The height of each shell is the difference between the upper curve and the lower curve, which can be expressed as h = 4/x³ - 1/2.
- The volume of each shell is given by V = 2πrh.
- Integrate the volume from x = 1 to x = infinity to get the total volume.
b) Revolving about the y-axis:
- The radius of each shell is the distance from the y-axis to the curve, which can be expressed as r = x.
- The height of each shell is the difference between the upper curve and the lower curve, which can be expressed as h = 4/x³ - 1/2.
- The volume of each shell is given by V = 2πrh.
- Integrate the volume from y = 1/2 to y = infinity to get the total volume.
c) Revolving about the line x=2:
- Translate the region so that the line x=2 becomes the new x-axis.
- Follow the same process as in part a) or b) to find the volume of the solid.
d) Revolving about the line y=4:
- Translate the region so that the line y=4 becomes the new y-axis.
- Follow the same process as in part a) or b) to find the volume of the solid.