Challenging +1.2 This is a parametric arc length question requiring the formula $\int \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$. While it involves differentiation, algebraic manipulation, and integration, the derivatives are straightforward, the expression under the square root simplifies nicely to a perfect square $(2t^2 - \frac{2}{t})^2$, and the final integration is routine. It's harder than average due to being Further Maths and requiring careful algebra, but the path is standard once the formula is applied.
1 The curve \(C\) is defined parametrically by
$$x = t ^ { 4 } - 4 \ln t , \quad y = 4 t ^ { 2 }$$
Show that the length of the arc of \(C\) from the point where \(t = 2\) to the point where \(t = 4\) is
$$240 + 4 \ln 2 .$$
1 The curve $C$ is defined parametrically by
$$x = t ^ { 4 } - 4 \ln t , \quad y = 4 t ^ { 2 }$$
Show that the length of the arc of $C$ from the point where $t = 2$ to the point where $t = 4$ is
$$240 + 4 \ln 2 .$$
\hfill \mbox{\textit{CAIE FP1 2008 Q1 [5]}}