Final answer:
To calculate the volume of a solid formed by rotating a region bounded by x = 3(y-7)² and x = 19 about the x-axis, solve for the intersection points to establish the limits of integration for y, then evaluate the integral of 2πrh across those limits.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves x = 3(y-7)² and x = 19 about the x-axis using the method of cylindrical shells, we need to integrate the volume of infinitesimally thin cylindrical shells across the bounded region.
First, we solve for the points of intersection by setting x = 3(y-7)² equal to x = 19, which gives us the limits of integration for y. Then, we set up the integral for the volume using the formula V = 2πrh, where r is the radius measured from the axis of rotation (the x-axis) to the curve, and h is the height of a shell, which equals the difference between x = 19 and x = 3(y-7)². The integral is then evaluated over the range of y values found from the intersection points.