To determine if your integral setup is correct, let's go through the process step by step.
The problem asks us to find the volume of the solid generated by revolving the region bounded by the graphs of y = x^2 and y = 4x - x^2 about the line x = 6.
To use the shell method, we need to consider disks or cylindrical shells that are oriented parallel to the axis of revolution. In this case, the axis of revolution is x = 6.
First, let's sketch the region bounded by the given curves to visualize it.
The curve y = x^2 is a parabola that opens upwards and passes through the origin. The curve y = 4x - x^2 is also a parabola, but it opens downwards and intersects the x-axis at (0, 0) and (4, 0). The region between these two curves is bounded by the x-values 0 and 4.
To set up the integral, we need to express the volume element as the product of the cylindrical shell's height, circumference, and thickness.
- Shell height: The difference in y-values between the curves y = 4x - x^2 and y = x^2 is (4x - x^2) - x^2 = 4x - 2x^2.
- Shell radius: The distance from the shell to the axis of revolution, x = 6, is 6 - x.
- Shell thickness: Since we want to think of the region as an infinite number of infinitely thin shells, the thickness will be dx.
The volume of each cylindrical shell is given by V = 2Ï€rhdx, where r is the shell radius, h is the shell height, and dx is the thickness.
Therefore, the integral to find the volume is:
V = ∫[0 to 4] 2π(6 - x)(4x - 2x^2) dx
Looking at your work, you correctly identified the shell radius as (6 - x) and the shell height as (4x - 2x^2). So your final integral:
V = 2π ∫[0 to 4] (6 - x)(4x - 2x^2) dx
represents the volume using the shell method for the given region and axis of revolution.
Now, you can proceed to evaluate the integral to find the exact volume by integrating the function and substituting the limits of integration.