Final answer:
To find the volume of the solid formed by rotating the region bounded by the given curves about the indicated axis of revolution, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid formed by rotating the region bounded by the given curves about the indicated axis of revolution, we can use the method of cylindrical shells. First, we need to find the points of intersection between the two curves. Setting y = -x^2 + 7 equal to x + y = 7, we get -x^2 + 7 = 7 - x. Solving this equation, we find that x = 3 and x = 4. Next, we need to find the height of each cylindrical shell by subtracting the lower curve from the upper curve. The height is (7 - x) - (-x^2 + 7) = x^2 - x. Finally, we integrate the volume of each cylindrical shell from 3 to 4. The volume of each shell is given by 2πrh, where r is the distance from the axis of revolution (x = 6) to the shell (which is 6 - x), and h is the height of the shell. Evaluating the integral, we find that the volume of the solid is approximately 4.19 cubic units.