Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y=1/x about the xx-axis, we can use the method of disks or washers. The volume of the solid is 2π/3.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y=1/x, y=1/x, y=0, y=0, x=1, x=1, and x=3x=3 about the xx-axis, we can use the method of disks or washers. Since the region is bounded by the curves y=1/x and y=0, we can integrate the areas of the disks or washers as we rotate the region about the xx-axis.
The area of each disk or washer is given by A=πr², where r is the radius of the disk or washer. In this case, the radius is the distance from the xx-axis to the curve y=1/x. Since the curve y=1/x intersects the xx-axis at x=1, the radius of each disk or washer is 1/x.
Therefore, the volume of the solid is given by V=∫[a,b]A(x)dx=∫[1,3]π(1/x)²dx=π∫[1,3]1/x²dx. Evaluating this integral, we get V=π(1/1-1/3)=π(2/3)=2π/3.