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3 [8 marks] Consider the integral | Vv 3a 2dz. 2 (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. 1 (a) [ e~*/2 (347 1) dx 0 (b) i (2x? 5) In(3x) dx [5 marks] Find the average value of the function f(x) = 2 e~*\" over x [0, L]. What happens to the average value as L - 00? [8 marks] A flower pot has shape given by rotating the graph of the function f(x) = 34+ Vz about the z-axis for x [0,25], where the units are cm and the base of the pot is at z = 0. (a) Find the volume of the flower pot, showing the working. (b) Suppose that the pot (standing on its base) contains soil up to height 20 cm, and 250 cm of fine mulch is added to the pot and levelled. Using f(20) and an appropriate cylindrical slice, estimate the height of the top of the mulch, showing the working. (Hint: /x increases quite slowly when is larger, therefore the top part of the pot is almost a cylinder.) . [16 marks] Solve the following differential equations. Note that the solution may be either a general solution or a particular solution. L cath (a) a =(y+1)%e73* with y(0)=2 (b) et 4ry+ \" =0 with y(1)=2 d (c) 3 + 2ty de = 0