Final answer:
The volume is 1,166,880.4 cubic units.
Step-by-step explanation:
To find the volume V generated by rotating the region bounded by the curves y = 12sqrt(x), y = 0, x = 1 about the axis x = -3, we can use the method of cylindrical shells 123.
The region bounded by the curves is shown below:
We can see that the axis of rotation is parallel to the y-axis and is located at x = -3. Therefore, we need to integrate concerning y.
The height of each cylindrical shell is given by the difference between the x-coordinate of the axis of rotation (-3) and the x-coordinate of the curve y = 12sqrt(x). Thus, the height of each shell is given by:
h = x + 3
The radius of each shell is given by the x-coordinate of the curve y = 12sqrt(x). Thus, the radius of each shell is given by:
r = y/2 = 6sqrt(x)
The limits of integration are y = 0 and y = 12. The integral expression for the volume V is given by:
V = 2π ∫[0,12] (x + 3)(6sqrt(x))^2 dx
Simplifying the expression, we get:
V = 72π ∫[0,12] x^(3/2) + 3x^(1/2) dx
Using the power rule of integration, we get:
V = 72π [(2/5)x^(5/2) + (6/7)x^(7/2)]_[0,12]
Evaluating the expression, we get:
V = 72π [(2/5)(12)^(5/2) + (6/7)(12)^(7/2)]
Therefore, the volume V generated by rotating the region bounded by the curves y = 12sqrt(x), y = 0, x = 1 about the axis x = -3 is approximately 1,166,880.4 cubic units.