Final answer:
To find the volume of solids revolved around lines y = 10, y = x², and y = 6x - x², use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of solids revolved around the lines y = 10, y = x², and y = 6x - x², we can use the method of cylindrical shells. First, we need to find the limits of integration by finding the x-coordinates where the curves intersect. Solving the equations x² = 6x - x² and x² = 10, we get x = 0, x = 2, and x = -2. Next, we need to find the height of each shell, which is the difference between the y-values of the curves at each x-coordinate. Finally, we integrate the function 2πx(y₁ - y₂) with respect to x from -2 to 2 to find the total volume of the solids.