Justification and explanation for each step will be appreciated

1. Multiple Choice [2i] pts] Circle the response that best answers each question. Each question is worth 4 points. There is no penalty for guessing and no partial credit. I. A spring has spring constant k = 2!] N f m. The amount of work required to stretch the spring 3m from its equilibrium position is: A.J 3.900.] 0.101).] D. dependent on whether the spring is stretched or compressed. E. None of the above I]. What is the mass of the wire from :L' = 0 to m = 5 whose density,r is given by: _ 32, 05x53 9(3) a, 31ch A. E units B. 45 units '3. 2'? units D. None of the above III. Which of the following integrals represents the area when the segment of the curve 3; = $2 from :r: = 1 to a: = 3 is revolved about the y-axis? s a s A. f 21ry1+4ndy E. f 21ry1+ydy U. f 27r31f1 +4rd3: 1 1 1 9 1 D. f 213\" 1' 1 + 43; dy E. None of the above 1 IV. The region in the ry-plane bounded by the lines y = r, r = 0 and y = 1 is the base of two different solids. The first solid has cross-sections perpendicular to the r-axis that are squares while the second solid has cross sections perpendicular to the r-axis that are semicircles. Let Vi be the volume of the first solid and V2 be the volume of the second solid. Which of the following best describes the relationship between Vi and V2? A. VI V2 D. Not enough information is provided to determine this. V. Let Wi be the amount of work required to stretch a spring 1 m from its equilibrium position and W2 be the amount of work required to stretch the spring an additional 1 m . Which of the following best describes the relationship between Wj and W2? A. WI W2 D. Not enough information is provided to determine this.2. Short Answer I. [5 pts] Find a function f [1:] such that f 1: f [as] do: = cos ($2) + C. Justify your response! 11. [Sp-ts] Let sbetheregionbouudedhyy=2+,y=3,e=2,mdx=s. When R is revolved about the line 1; 2 1I a solid of revolution is formed. This solid can be approximated by slicing the region into 3 rectangles of equal width that coincide with the functions at their lefthend endpoints, then revolving those rectangles about the line 3; = 1. Find the volume of the shape obtained by revolving the shaded rectangle about the line 3; = 1. 3. [20 pts] The region R is hounded by the curves y = :t:2 and y = E 3:, which are shown below: I. Set up, but do not evaluate, an integral or a sum of integrals with respect to a: that would give the area of R. 11. Set up, but do not evaluate, an integral or a sum of integrals with respect to :9! that would give the area of R. III. Suppose now that the region R is revolved about the line 3 = l3. Set up, but do not evaluate, an integral or a sum of integrals that would give the volume of the resulting solid. d. [15 pts] Find the length of the segment of the curve y = %[2 + 1'2)\": from I = 0 t0 1: = 1. Simplify your answer completely! 5. [20 pts] Evaluate the following antiderivatives. 1. / Ar + 6 4x2 + 1 - dx II. evi da Hint: First, make the substitution w = vi.5. [15 pts] The magnitude of the electric force between two particles with charge ql and 92 is given by Coulomb's Law: F = kfi'z r2 where k is a constant and r is the distance between the two particles. Suppose now that there is a thin1 rod that extends from 1: = [J to :r = 2 with total charge Q and that this charge is distributed over the rod uniformly. Now, a particle with charge :3 units is placed at :r = 3. Follow the steps below to compute the total force that the rod exerts on the particle. x hO'l _ x 2 3 Coulomb's law cannot be applied directly because the magnitude of the force exerted by different segments of the rod on the particle are dierentl Thus, the rod cannot be treated as a particle! However, the force exerted by a small segment of the rod of length has, centered around a value 1: between 1: = I] and r = 2 can be approximated by treating the small segment as a particle. By treating the small segment above as if it were a particle: I. Write an expression for the total charge Q contained in the segment of length :13 [You may call the charge {HQ and the segment length d3 if you prefer}. 11. Write an expression for the approximate force F that the small segment exerts on the particle of charge 9' at a = 3 {You may call the force dF if you prefer}. [11. Write down an integral that gives the total force, F, that the rod exerts on the particle