Final answer:
To find the volume generated by rotating the region bounded by the curves y = 5x - x² and y = 4 about the axis x = 1, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume generated by rotating the region bounded by the curves y = 5x - x² and y = 4 about the axis x = 1, we can use the method of cylindrical shells.
Step 1: First, we need to find the limits of integration by setting the two equations equal to each other: 5x - x² = 4. Then solve for x to find the x-values where the curves intersect.
Step 2: Next, we need to find the height of each cylindrical shell. This can be done by subtracting the y-values of the curves at each x-value.
Step 3: Finally, we can use the formula for the volume of a cylindrical shell to calculate the volume generated by each shell. The formula is V = 2πrh, where r is the distance from the axis of rotation to the shell and h is the height of the shell.
Step 4: Integrate the volumes of all the shells from the minimum x-value to the maximum x-value to find the total volume generated.