Final answer:
The volume of a solid of revolution using the shell method is obtained by integrating the volume of each shell, with the formula dV = 2πx (√x−2 +1 – (x−1)) dx, across the bounds of x. The volume of a sphere is given by the formula ⅔πr³.
Step-by-step explanation:
The question relates to the volume of a solid of revolution created by rotating the region bounded by the graphs of x−1 and √x−2 +1, around the x-axis. To find the volume using the shell method, we consider a cylindrical shell at distance x from the y-axis with height as the difference of the functions, which is √x−2 +1 – (x−1). The volume dV of one thin shell with a thickness dx is given by the circumference of the shell 2πx, times the height of the shell, times the thickness of the shell.
The formula to find the volume
V
of one shell is:
dV = 2πx (√x−2 +1 – (x−1)) dx
The total volume V of the solid is then obtained by integrating this expression for dV from the lower to the upper bound of x where the two functions intersect.
The volume of a sphere is ⅔πr³, not ⅔πr². The latter represents the surface area of a sphere. In terms of dimensional analysis, only formulas with dimensional consistency like V = πr²h (volume of a cylinder) are valid.