MATH 1241: Calculus II Assignment 2 (100 marks total) Note: Give exact answers where possible (for example, è is exact but 3.142 is approximate). 1. [10 marks] Find the area

of the region bounded between the curves y = x and y = 2 - x² by: a. (4 marks) Integrating with respect to x b. (6 marks) Integrating with respect to y 2. [10 marks] Find the area of the region(s) enclosed by the curves y = x² and y = x³ – 2x. Give the exact value. A2-1 3. [12 marks] Use the indicated method to find the volume of the solid generated when the region between the curves y = (x - 1)² and y = 1 is rotated around the indicated axis: a. (6 points) The x axis (use the method of discs and washers) b. (6 points) The y axis (use the method of cylindrical shells) 4. [10 marks] Consider the region enclosed by the curves y = √x and y = x. Find the volume of the solid generated when the region is revolved about the line x = 2. TRU Open Learning A2-2 5. [10 marks] Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves y = x² and x = y² about the line x = −1. 6. [10 marks] = The base of a solid is the region bounded by the parabolas y x² and y = 2x². Find the volume of the solid if the cross sections perpendicular to the x-axis are equilateral triangles. 7. [10 marks] Find the volume of the solid S if the base of S is the triangular region with vertices (0,0), (3,0), and (0,2) and cross sections perpendicular to y-axis are semi- circles. 8. [8 marks] A log 10 m long is cut at 1-metre intervals and its cross-sectional areas (at a distance x from the end of the log) are given in the table. Use the Midpoint Rule with n = 5 to estimate the volume of the log. 0 01 2 x(m) A(m²) 0.68 0.65 0.64 3 4 5 0.61 0.58 0.59 Assignment 2 6 7 TRU Open Learning 8 9 10 0.53 0.55 0.52 0.5 0.48 9. [12 marks] A conical tank rests on a platform that is 10 metres above a large lake. The circular base of the tank has a radius of 4 metres and the top of the tank is 15 metres above the surface of the lake. How much work must be done to fill the tank with water from the lake if it is to be pumped in through an outlet at the bottom of the tank? Set up your coordinate system so that x = 0 corresponds to the top of the tank and x = 5 corresponds to the base of the tank, which rests on the platform. Use the fact that the density of water is 1000 kg/m³ and g = 9.8 m/s². MATH 1241: Calculus II 10. [8 marks] If a cup of coffee has temperature 95 °C and is placed in a room where the temperature is 20 °C, then the temperature T of the coffee after t minutes is A2-3 t modelled by T(t) = 20 + 75e ¯50. What is the average temperature of the coffee during the first half hour? Select resources associated with the following course textbook are used with permission from Cengage Learning: Stewart, J. (2016). Single variable calculus: Early transcendentals (8th ed.). Boston, MA: Cengage Learning. TRU Open Learning