5. Suppose that a function f satisfies the following conditions: o f is positive and continuous on (0,00). f'(x) 0. The improper integral $ * = f(x) dx converges Let R be the region between the graph of y = f(x) and the r-axis for r > 0, and let V(a) be the volume of the solid generated by revolving the region R about the line I = a where a is a nonnegative real number. . Show that the improper integral f(2) d.c converges. b. Express V(a) using the cylindrical shells method. c. Show that V"(a) > 0 for all a > 0. 5. Suppose that a function f satisfies the following conditions: o f is positive and continuous on (0,00). f'(x) 0. The improper integral $ * = f(x) dx converges Let R be the region between the graph of y = f(x) and the r-axis for r > 0, and let V(a) be the volume of the solid generated by revolving the region R about the line I = a where a is a nonnegative real number. . Show that the improper integral f(2) d.c converges. b. Express V(a) using the cylindrical shells method. c. Show that V"(a) > 0 for all a > 0