Given that A is constant and the integral from 1 to 4 of (3 sqrt(x) + A) dx = 2A^2, show that there are exactly two possible values for A. (5)

(a) Express the limit, as dx tends to 0, of the sum from x = 2.1 to 6.3 of (2/x) dx, as an integral. (1) (b) Hence show that this limit equals ln k, where k is a constant to be found. (2)

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 3 shows a sketch of part of a curve with equation y = ((x - 2)(x - 4)) / (4 sqrt(x)), x > 0. The region R, shown shaded in Figure 3, is bounded by the curve and the x-axis. Find the exact area of R, writing your answer in the form a sqrt(2) + b, where a and b are constants to be found. (6)