The region T in the plane is bounded on the left by x=y2 and on the right by x=6−y2. The line x=2 intersects the interior of T. The region T is revolved about the line x=2 to create a solid.
(a) Find the y-coordinates of the intersection points of the bounding curves.
(b) Determine analytically the value(s) of y at which the distance from the line x=2 to x=y2 equals the distance from the line x=2 to x=6−y2. Justify your work algebraically.
(c) Write a piecewise definite integral (or a sum of definite integrals) that exactly represents the volume of the solid. Label clearly the limits associated with each integral.
(d) Evaluate the integral(s) obtained in part (c) to determine the exact volume.