Final answer:
To calculate the volume of the solid formed by rotating the region bounded by y = x³, y = 0, and x = 4 about y = 8.8, we use the method of cylindrical shells, integrating from x = 0 to x = 4, and accounting for the subtraction of the shell's radius from the axis of rotation.
Step-by-step explanation:
Calculating Volume Using Cylindrical Shells
To find the volume V generated by rotating the region bounded by y = x³, y = 0, and x = 4 about y = 8.8, we use the method of cylindrical shells. The volume is found by integrating the circumferential area of each cylindrical shell from the lower to the upper bound of x, which are 0 and 4, respectively.
The radius of each shell is (8.8 - y), and the height of the shell is x. The formula for the volume of a cylinder, V = πr²h, is relevant because a cylindrical shell is like a cylinder with a very small thickness, Δx.
The general formula using cylindrical shells is given by V = 2π ∫ (radius)(height)(thickness). For this problem specifically, we have V = 2π ∫_{0}^{4} (8.8 - x³)(x)(dx). Calculating this integral will give us the volume of the solid.