Challenging +1.2 This is a straightforward application of the polar area formula requiring integration of r² with respect to θ, followed by solving an equation involving exponentials and logarithms. While it involves Further Maths content (polar coordinates) and requires careful algebraic manipulation, the method is direct with no conceptual surprises—students simply apply the standard formula A = ½∫r²dθ and solve the resulting equation.
\includegraphics{figure_1}
The curve \(C\) has polar equation
$$r = \theta^{\frac{1}{2}}e^{\theta/\pi},$$
where \(0 \leq \theta \leq \pi\). The area of the finite region bounded by \(C\) and the line \(\theta = \beta\) is \(\pi\) (see diagram). Show that
$$\beta = (\pi \ln 3)^{\frac{1}{2}}.$$ [6]
\includegraphics{figure_1}
The curve $C$ has polar equation
$$r = \theta^{\frac{1}{2}}e^{\theta/\pi},$$
where $0 \leq \theta \leq \pi$. The area of the finite region bounded by $C$ and the line $\theta = \beta$ is $\pi$ (see diagram). Show that
$$\beta = (\pi \ln 3)^{\frac{1}{2}}.$$ [6]
\hfill \mbox{\textit{CAIE FP1 2003 Q1 [6]}}