please show all working out as i want to compare it to my work thank you

3 1. [3 marks] Consider tlle integral f 1.1\" 31' 2 (Ir. 2 (a) Using the trapezoidal rule with 4 subinter'walsf estimate the integral numerically. (1)) 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. 2. [8 marks] Find the following integrals. Show all working. 1 (a) f c 5:2{3122 1) d1: . a (b) I/[Z'cz 5) ln{:3:1:]d:1: 3. [5 marks] Find the average value of the function 31:12) 2 2 c 4" over :1: E [0. L]. 'What happens to the average value as L ) De? 4. [3 marks] A ower pot has shape given 11}; rotating the graph of the function f[:1:j = 3 + V'G about the .r-axis for :1: E [0,25]. where the units are em and the base of the pot is at :1: = 0. (a) Find the volume of the ower potf showing the working. (1)) Suppose that the pot {shunting on its base} contains soil up to height 20 cm. and 250 cm3 of ne muld1 is added to the pot and levelled. Using 3120} and an appropriate cylindrical slice, estimate the height of the top of the mulch, showing the working. (Hint: VIE increases quite slowlyr when :1: is larger. therefore the top part of the pot is almost a cylinder.) 5. [16 marks] Solve the following differential equations. Note that the solution may be either a general solution or a particular solution. d (a) l 2 {31+ 1]!2 r: h with 31(0) 2 2 1'. d 1 (b) 322?: 4mg + I = [l with 331(1) = 2 (c) 33: + $33,: 132:{ 2 0