2. (a) Writhout using a calculator, determine the following integral: 133243:+9 /_a. n $2+43+25 (Hint: First write the integrand H33) as I _3:243:+9_1+ (13+?) (I)_x2+4$+25_ $2+4x+25 where e and b are to be determined.) (b) A stained glass window is constructed from two panes of glass: a rectangular pane of clear glass and a semicircular pane of coloured glass (see gure]. The height of the clear pane is h and the radius of the coloured pane is r. g h, | | | | I I | | | I I I I I I | | | I I I I 1r (i) write down an expression for the area AL of the clear pane (in terms of h and r). (ii) write down an expression for the area A2 of the coloured pane (in terms of h and I\"). (iii) \"trite an expression for the perimeter P of the entire window (in terms of h, and 1'). (iv) Assume that the amount of light entering through the clear pane is ll]: DUBAI units where A1 is the area of the clear pane, and that the amount of light entering the through the coloured pane is 5, 000A; units where A2 is the area of the coloured pane. \"trite an expression for the amount of light L entering through the entire window in terms of h and r. (v) The perimeter is required to be 8 metres. Use your answer for (iii) to express h as a function of r, and so write the function L (from (iv) ) as a function of r (that is, eliminate h). (vi) The client requires L = 20, 000 units to enter through the window. Use Newton's method with an initial guess of ro = 0.3 to find an approximation of the value of r (accurate to 3 decimal places) which produces L = 20, 000. (Newton's method for solving f(r) = 0: f (In) for n = 0, 1, 2, ... ) f' (rn) (vii) Explain how you can check your approximation of r