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Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis. y = 12sqrt(x) ,y = 0, x = 1; about x = −3

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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.

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