(a) Without using a calculator, determine the following integral: x²8x+52 So 2² +82 +52 dx. (Hint: First write the integrand I(x) as 2²-8x+52 I(x) = = 1+ ax+b x² + 8x +52 x² + 8x + 52 where a and b are to be determined.) (b) A steel storage tank for propane gas is to be constructed in the shape of a right circular cylinder with a hemisphere at each end. Suppose the cylinder has length l metres and radius r metres. (i) Write down an expression for the volume V of the storage tank (in terms of l and r). (ii) Write down an expression for the surface area A of the storage tank (in terms of land r). (iii) Using the result of part (ii), write V as a function of r and A. (That is, eliminate (.) (iv) A client has ordered a tank, but can only afford a tank with a surface area of A = 40 square metres. Given this constraint, write V = V(r). (v) The client requires the tank to have volume V = 10 cubic metres. Use Newton's method, with an initial guess of ro = 2 to find an approximation (accurate to three decimal places) to value of r which produces a volume of 10 cubic metres. (Newton's method for solving f(r) = 0: f(rn) Tn+1 = Tn - for n= 0, 1, 2,...) f'(rn)