Question

Use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by y=1/x, x=1/2, x=4 and the x-axis about the y-axis.

Answer 1

To find the volume of the solid formed by rotating the given region around the y-axis, we use the method of cylindrical shells. We integrate from x=1/2 to x=4 to determine the volume as 7π cubic units.

Step-by-step explanation:

To determine the volume of the solid obtained by rotating the region bounded by the curve y=1/x, the lines x=1/2 and x=4, and the x-axis about the y-axis, we use the method of cylindrical shells. The volume of a thin cylindrical shell with radius r and height h is given by the circumference of the circle times the height times the thickness, V = 2πrhΔr. Here, height of a shell at any x is given by the function y=1/x.

Integrating from x=1/2 to x=4, we have:

• The integral setup: V = ∫ from 1/2 to 4 (2πx)(1/x)dx.
• Simplifying, we get: V = 2π∫ from 1/2 to 4 dx.
• After integrating, we find: V = 2π[x] evaluated from 1/2 to 4.
• Volume of the solid: V = 2π(4 - 1/2) = 7π cubic units.

Thus, the volume of this solid is 7π cubic units.