A region S in the first quadrant is bounded on the left by the y–axis and on the right by the vertical line x=3. Its upper boundary is given by the piecewise linear function g(x) whose values are shown in the table below. The lower boundary is the x–axis.
x (0) 1 2 3
g(x) 0 2 1 3
Between the tabulated points, g is linear on each sub-interval. The region S is revolved about the vertical line x=5 to generate a solid.
(a) Using the data in the table, write a definite integral (or sum of definite integrals) whose value equals the volume of the solid. Clearly indicate the integrand on each sub-interval.
(b) Evaluate the integral(s) exactly to obtain the volume.