Question

Consider the solid obtained by rotating the region bounded by the given curves about the x-axis

y=1/x, x=1, x=5, y=0
Find the volume V of the solid.

Answer 1

The volume V of the solid obtained by rotating the region bounded by the curves y=1/x, x=1, x=5, and y=0 about the x-axis is V = 4π/5 cubic units.

Step-by-step explanation:

To find the volume V of the solid obtained by rotating the region bounded by the curves y=1/x, x=1, x=5, and y=0 about the x-axis, we can use the method of cylindrical shells or the disk method. In this case, the disk method is more suitable. The formula for the volume of a solid of revolution using the disk method is ℝ V = ∫_a^b π • (radius of disk)^2 • dx, where in this problem, the radius of the disk is given by y=1/x, a=1, and b=5.

By integrating, we get the volume V of the solid:
V = ∫_1^5 π • (1/x)^2 • dx
= π ∫_1^5 1/x^2 • dx
= π [-1/x]_1^5
= π (-1/5 + 1/1)
= π (4/5)

Therefore, the volume is V = 4π/5 cubic units.