(b) Since .
(c) When S is rotated about the x-axis, its volume can be obtained using
disks:See the figure, where R is the point (a, b), and seek a such thatGraph y = sin 2x in [−1, 3.2] × [−1, 1]. Note that y = f ′′.(a) The graph of f is concave downward where f ′′ is negative, namely, on
(b, d). Use the calculator to solve sin 2x = 0, obtaining b = 1.651 and d =
2.651. The answer to (a) is therefore 1.651 < x < 2.651.