Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 1/9x², x = 5, and y = 0 about the y-axis, you need to use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 1/9x², x = 5, y = 0 about the y-axis, we need to use the method of cylindrical shells. The volume of each shell is given by the formula V = 2πx(f(x) - g(x)) dx, where f(x) is the upper curve, g(x) is the lower curve, and x represents the variable along the axis of rotation.
In this case, the upper curve is y = 1/9x² and the lower curve is y = 0. The axis of rotation is the y-axis. So, we can calculate the volume as follows: V = ∫[0,5] 2πx(1/9x² - 0) dx.
Simplifying the integral, we get V = 2π/9 ∫[0,5] x³ dx = 2π/9 [x⁴/4] [0,5] = (π/18)(5⁴ - 0⁴) = 625π/18.