Hello, please help me solve all of the questions attached. And please show all necessary work.

1. Consider the region in the first quadrant that is enclosed by the functions y = x , y = x + 2, and x = 0. For each of the following, set up the definite integral (s) that would solve the question, but don't simplify it at all, and don't evaluate it either! (but know how to do that, of course, for the test) a) Find the area of the region by integrating with respect to x. Answer: Area = top function - bottom function A = So [(x +2) -(x2) ]dx 4 -3 (Note: Answer given for a) to show you how much work we're looking for on these questions. b) Find the area of the region by integrating with respect to y. c) The region is rotated around the x axis - find the volume by the method of discs. d) The region is rotated around the y axis - find the volume by the method of shells. e) The region is rotated around the line y = -2. Find its volume using discs. f) The region is rotated around the line x = -4. Find its volume using shells. Page 12. When a 1.0 N force is applied to a spring, it is 3. What is the average value of f(x) = ex - e-* on stretched 0.25 m from its resting position. Calculate the interval [1, 3]? (Round to 2 dec. places.) the work done on the spring to stretch it from its resting position to that distance. 4. A thick 10 m rope that weighs a total of 200 N is hanging over the edge of a cliff. How much work does it take to lift the entire rope up? (Try this, and if you get stuck, see the HINT in the Answers below.)5. A car which begins at position d = 0, travels according to the following velocity function: v = 10t - 4t'm/s. At the same time, at a position of d = 5 m, a trucks begins at rest, and travels at a constant acceleration, a = 6 m/s. a) Find the acceleration function and the position function of d) What's the distance the car travels in the the car. first 3 seconds? (show the integral set up first, and then use the graph. calc. to solve) b) Find the velocity function and the position function of the truck. e) How far apart are the two vehicles after 3 sec? (solve algebraically) c) What's the car's displacement in the first 3 seconds? (solve algebraically)