(a) For each of the following integrals, determine if they are proper or improper. If an...

 

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(a) For each of the following integrals, determine if they are proper or improper. If an integral is improper, briefly explain why (one sentence), and write it as a limit (or sum of limits) of proper integrals. (i) e-2x dx C/2 1 dx (iii) cos(x) L 2x2 dx (x+3)3 (b) [2 points] Determine whether the improper integral cos(x) dx is convergent or divergent. Use the definition seen in class to explain why. Q6. Consider the curves y = -2x(x 2) and y = x. - (a) [1 point] Find the x-values of the points of intersection of these curves. Draw a diagram to represent the curves and these x-values. Show your work. (Note: if you can't find these points, for the following parts, just represent them by a and b, where a < b.) (b) [2 points] Compute the area of the region R enclosed by these two curves. Show your work. (c) [3 points] Consider the solid formed by revolving the region R (of part (b)) around the axis y = 1. Give a definite integral that computes the volume of this solid by the washer method. (Do not evaluate the integral.) Show your work. (d) [3 points] Consider the solid formed by revolving the region R (of part (b)) around the axis x = 0. Give a definite integral that computes the volume of this solid by the shell method. (Do not evaluate the integral.) Show your work. Q7. Consider a tank with the following shape and dimensions. x (meters) 2 0 10m 4m (a) [3 points] Suppose the tank is filled with water. Give a definite integral that computes the total work (in Joules) required to lift all of the water to 2m above the top of the tank. (Do not evaluate the integral.) Show your work. Note that the density of water is 1000kg/m and the acceleration due to gravity is 9.8m/s. (b) [1 point] What would the definite integral be if the tank were only filled halfway to the top? You can give the integral without justification. (a) For each of the following integrals, determine if they are proper or improper. If an integral is improper, briefly explain why (one sentence), and write it as a limit (or sum of limits) of proper integrals. (i) e-2x dx C/2 1 dx (iii) cos(x) L 2x2 dx (x+3)3 (b) [2 points] Determine whether the improper integral cos(x) dx is convergent or divergent. Use the definition seen in class to explain why. Q6. Consider the curves y = -2x(x 2) and y = x. - (a) [1 point] Find the x-values of the points of intersection of these curves. Draw a diagram to represent the curves and these x-values. Show your work. (Note: if you can't find these points, for the following parts, just represent them by a and b, where a < b.) (b) [2 points] Compute the area of the region R enclosed by these two curves. Show your work. (c) [3 points] Consider the solid formed by revolving the region R (of part (b)) around the axis y = 1. Give a definite integral that computes the volume of this solid by the washer method. (Do not evaluate the integral.) Show your work. (d) [3 points] Consider the solid formed by revolving the region R (of part (b)) around the axis x = 0. Give a definite integral that computes the volume of this solid by the shell method. (Do not evaluate the integral.) Show your work. Q7. Consider a tank with the following shape and dimensions. x (meters) 2 0 10m 4m (a) [3 points] Suppose the tank is filled with water. Give a definite integral that computes the total work (in Joules) required to lift all of the water to 2m above the top of the tank. (Do not evaluate the integral.) Show your work. Note that the density of water is 1000kg/m and the acceleration due to gravity is 9.8m/s. (b) [1 point] What would the definite integral be if the tank were only filled halfway to the top? You can give the integral without justification.