Final answer:
To find the volume of the solid obtained by rotating a region bounded by a circle about a given axis, we can use the method of cylindrical shells. In this case, the volume of the solid obtained by rotating the region R, bounded by the circle x^2 + y^2 = -1, about the axis y = -5 is 8π.
Step-by-step explanation:
The region R is bounded by the circle with equation x2 + y2 = -1. To find the volume of the solid obtained by rotating R about the axis y = -5, we can use the method of cylindrical shells. The volume of each shell is given by the product of the height of the shell (which is the circumference of the circle at that particular height) and the differential thickness of the shell (which is dx).
The circumference of the circle is 2πr, where r is the radius of the circle at that particular height. From the equation x2 + y2 = -1, we can see that the radius of the circle is the square root of 1, which is 1. Therefore, the circumference is 2π.
Since we are rotating the region R about y = -5, this means that the height of each shell will be the difference between the y-coordinate of the top of the shell (which is -5) and the y-coordinate of the bottom of the shell (which is -1). Therefore, the height of each shell is 4.
By integrating the product of the circumference and height over the range of x = -1 to x = 1, we can find the total volume of the solid:
V = ∫[from -1 to 1] 2π(2) dx = 4π ∫[from -1 to 1] dx = 4π(x) [from -1 to 1] = 4π(1 - (-1)) = 8π.