1. Shell Method; Consider the region between f(x) = x - 2 and the x-axis on [2, 5]. For each line of rotation given, use the shell method to construct definite integrals to find the volume of the resulting solid. Around the y-axis Around the line y = -2 6-4-2 -2 2 4 6 8 10 -4+ - et -et (a) (d) Around the x-axis Around the line x = 2 - 2, 4-2, 14 6 8 10 (b) (e) Around the line y = 3 Around the line r = 6 -32 4 6 8 10 12 -4+ -6- (c) (f) 2. Arc Length; Each definite integral in the following Exercises represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval. 3 (a) V1 + 9ebr dx (c) -2 (b ) VI + sin?(x)dx (d) VI + 36:de3. Arc Length; Find the exact value of the arc length of each function f(x) in the following exercises on [a, b] by writing the arc length as a definite integral and then solving that integral. (a) f(x) = 3x + 1, [a, b] = [-1, 4] (b) f(x) = 4-1, [a, b] = [2, 5] 4. Suppose a drinking fountain expels water so that it falls in the shape of the parabola f(x) = 3 -$o', measured in inches, as shown in the given figure. Set up and solve a definite integral that measures the distance that water travels starting from when it leaves the fountain (at x = 0) and ending when it hits the surface of the fountain (when f(x) = 0). Path of water from a drinking fountain 2 5. Surface Area; set up and solve a definite integral to find the exact area of each surface of revolution obtained by revolving the curve y = f(2) around the x-axis on the interval [a, b]. (a) f(x) = x, [a, b] = [0, 2] (b) f(x) = 1, [a, b] = [1, 10] (c) f(x) = VI, [a, b] = [0, 4] (d) f(x) = e-*, [a, b] = [-1, 1]