Problem

Figure 1

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of the curve C defined by the parametric equations

$$x = (2t+3)^{3/2} \qquad y = \frac{3}{2}t^2 + 3t + 6 \qquad -\frac{3}{2} \le t \le 3$$

(a) Show that

$$\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = a(t+2)^2$$

where $a$ is an integer to be determined.

Hence, using algebraic integration, determine

(b) the exact length of C,

(c) the exact area of the surface generated when C is rotated through 360º about the x-axis, giving your answer in the form $k\pi$ where $k$ is a rational number.