Final answer:
To find the volume of the solid obtained by rotating the region bounded by given curves around the line x = -1, we use the shell method in integral calculus and integrate the volume of cylindrical shells from x = 1 to x = 4, giving us a volume of 2π (3 + ln(4)).
Step-by-step explanation:
The question asks for the volume of the solid obtained by rotating the region bounded by xy = 1, y = 0, x = 1, and x = 4 about the line x = -1. To find this volume, we could use the shell method in integral calculus given the nature of rotation about a line not on the boundary of the region.
First, we need to express y in terms of x from the equation xy = 1, which gives us y = 1/x. Then, the volume of the solid is given by the integral of 2π (radius)(height) from x = 1 to x = 4. The radius is the distance from each slice to the line x = -1, so the radius is x + 1. The height is the value of the function, which is 1/x. Therefore, the volume integral becomes V = 2π ∫ (x + 1)(1/x) dx from x = 1 to x = 4.
Executing the integration yields:
V = 2π [∫ (1) dx + ∫ (1/x) dx] from x = 1 to x = 4 = 2π [x + ln|x|] from 1 to 4.
Plugging in the limits, we find the volume V = 2π [4 + ln(4) - (1 + ln(1))] = 2π [4 + ln(4) - 1], since ln(1) is zero.
Finally, we simplify this to get:
V = 2π (3 + ln(4)). This is the volume of the solid when the given region is rotated around the line x = -1.