[8 marks] Consider the integral J9 4x dz. 1 (a) Using the trapezoidal rule with 4 subintervals, estimate the integral numerically. (b) Find the integral exactly (using calculus and showing the working) and compare this with your numerical result by calculating the absolute value of the error. [8 marks] Find the following integrals. Show all working. (a) e 2x (x2 - 3) dx (b) / (1 + 3x2 ) In(2x) dx\f[8 marks] The shape of a 20-cm tall flower vase is given by rotating a curve about the i z-axis for x [0,20]. This curve consists of the graph of f(z) = 5+ avr for x [0,16] 1 and the graph of g(z) = g(t 16)* + 7 for x [16,20]. The units are cm and the base of the vase is at x = 0. (a) Show the shape of the vase by sketching the graphs of f(x) and g(x) in the respective intervals given above. What is the radius at the top of the vase? What is the base radius? (b) Find the volume of the vase, showing the working. [16 marks] Solve the following differential equations. Note that the solution may be either a general solution or a particular solution. d (a) 3a = (y+ 2)? (22 3r) with y(0)=1 (b) 209 62*y+2*=0 with y(1)=4 c dy (c) a7 + x27y 327 =0 a