Challenging +1.2 This is a standard parametric surface area of revolution problem requiring the formula S = 2π∫y√((dx/dt)² + (dy/dt)²)dt. While it involves several steps (finding derivatives, substituting into the formula, simplifying the integrand to get √(1+4t²), and integrating), the techniques are routine for Further Maths students and the question provides the target answer to work towards, reducing problem-solving demand.
1 A curve is defined parametrically by
$$x = a t ^ { 2 } , \quad y = a t$$
where \(a\) is a positive constant. The part of the curve joining the point where \(t = 0\) to the point where \(t = \sqrt { } 2\) is rotated through one complete revolution about the \(x\)-axis. Show that the area of the surface obtained is \(\frac { 13 } { 3 } \pi a ^ { 2 }\).
1 A curve is defined parametrically by
$$x = a t ^ { 2 } , \quad y = a t$$
where $a$ is a positive constant. The part of the curve joining the point where $t = 0$ to the point where $t = \sqrt { } 2$ is rotated through one complete revolution about the $x$-axis. Show that the area of the surface obtained is $\frac { 13 } { 3 } \pi a ^ { 2 }$.
\hfill \mbox{\textit{CAIE FP1 2007 Q1 [4]}}