Give examples of functions, which satisfy the following conditions, and justify your choice. If no such functions exists, explain why. a. A function f(r) such that f f(r) dr converges, but f(x) dz diverges. b. A function f(r) such that both f(z) dr and f(x) dx diverge. c. A function f(r), such that 0≤ f(r) ≤ 10 for every z E [0, 0o) and f f(x) da diverges. d. A function f(z), such that 0≤ f(r) ≤ 10 for every z € (0,00) and (r) da converges. 3. Find the area of the region bounded by the curves y = √ and y = -2² between 2 = 0 and 2 = 1. Show your steps. (Hint: Juse slicing.) 4. The region bounded by the curves y = x² and y=z³ is rotated around a. the r-axis: b. the y-axis c. the line y=1. Compute the volume of the resulting solid in each of the three cases. Show your steps. 5. The fictional city of Torontino radially has a population density of 1000e-001 people per km², where r is the radius (in km) from QM tower. What is the total population living within 5 km of the QM tower? Show your steps. 6. Consider the Initial Value Problem (IVP) given by dy da (0)=1. a. Sketch the slope field of the differential equation for values of (x, y) in the square (-1, 1) × (-1, 1). You can use digital graphing tool if needed; however, verify the slope field by computing the slope at, at least, two points. b. Use the Euler method to approximate the solution (2) of the IVP at points 2 = 0.0.1. 22 0.2, 23=0.3. Show your steps. c. Find the exact solution of the IVP. Show your steps