Please answer the entirety of this question. Focusing on sections (v), (vi) and (vii), as these are the main sections not understood. Thank you.

2. (a) Without using a calculator, determine the following integral: - Ar + 9 1 4 41 + 25 dr. (Hint: First write the integrand I(x) as I' - 4r +9 at + b I(I) = r' + 4x + 25 -1+ 12 + 4r + 25 where a 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 semi-circular pane of coloured glass (see figure). The height of the clear pane is h and the radius of the coloured pane is r. (i) Write down an expression for the arca A, of the clear pane (in terms of h and r). (ii) Write down an expression for the area As of the coloured pane (in terms of h and r). (iii) Write an expression for the perimeter P of the entire window (in terms of h and r)- (iv) Assume that the amount of light entering through the clear pane is 10, 000A, units where A, is the area of the clear pane, and that the amount of light entering the through the coloured pane is 5,000Ag units where A, is the arca of the coloured panc. Write an expression for the amount of light & 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 " (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 ( T m) f' (rm ) for n = 0, 1, 2, ...) (vii) Explain how you can check your approximation of r