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=8-x^2, y= x^2 ; about x=2
V=?
To find the volume of the region generated by rotating the given curves about the specified axis using the method of cylindrical shells, follow these steps:
Step 1: Draw a sketch of the given curves and the axis of rotation.
In this case, the given curves are y = 8 - x^2 and y = x^2, and the axis of rotation is x = 2.
Step 2: Determine the limits of integration.
Since the axis of rotation is x = 2, the intervals of integration will be from x = 0 to x = 2 (the two curves intersect at x = 0), as we need to consider the region to the left of the axis of rotation.
Step 3: Express the height of the cylindrical shells.
The height of each cylindrical shell will be the difference between the y-values of the two curves at a given x-value. In this case, the height will be (8 - x^2) - (x^2), which simplifies to 8 - 2x^2.
Step 4: Establish the radius of the cylindrical shells.
The radius will be the distance from the axis of rotation (x = 2) to a given x-value. Therefore, the radius will be (2 - x).
Step 5: Set up the integral to compute the volume.
The volume V will be given by the integral of the product of the height and the circumference of the cylindrical shells:
V = ∫[0 to 2] 2π(2 - x)(8 - 2x^2) dx
Step 6: Evaluate the integral to find the volume.
You can now calculate the integral using the limits of integration determined in Step 2. Solve the integral to find the value of V.
V = 2π ∫[0 to 2] (16 - 4x^2 - 16x + 2x^3) dx
V = 2π [16x - (4/3)x^3 - 8x^2 + (1/2)x^4] evaluated from 0 to 2
V = 2π [(32 - (4/3)(8) - 32 + (1/2)(16)) - (0 - (4/3)(0) - 0 + (1/2)(0))]
V = 2π [(32 - (32/3) - 32 + 8) - (0)]
V = 2π [(8/3)]
V = (16π/3)
Therefore, the volume generated by rotating the region bounded by the given curves about the axis x = 2 is (16π/3) cubic units.