Explanation:
The Shell Method can be used to find the volume of the solid generated by revolving the region bounded by the curves y = √x, y = 0, y = x - 2 about the x-axis.We need to follow these steps to solve the problem using the Shell Method:1. First, we need to sketch the region bounded by the curves and the axis of revolution.2. Next, we need to determine whether to use the Shell Method or the Disk Method. Since we are revolving the region around the x-axis, we will use the Shell Method.3. Now, we need to express the curves in terms of x instead of y. y = √x can be written as x = y^2 and y = x - 2 can be written as x = y + 2.4. We can now set up the integral using the formula for the Shell Method:∫(2πrh)dxwhere r is the distance from the axis of revolution to the shell, and h is the height of the shell.5. In this case, the radius of the shell is x, and the height is (x - √x). Therefore, the integral is:∫(2πx(x - √x))dx from x = 0 to x = 46. Integrating this expression, we get:V = 2π[(1/3)x^3 - (2/5)x^(5/2)] from x = 0 to x = 4V = 32π/5 - 8π/3The volume of the solid generated by revolving the region bounded by the curves y = √x, y = 0, y = x - 2 about the x-axis is 32π/5 - 8π/3.
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