曲线 $C$ 的极坐标方程为 $r^2=\sin2\theta\cos\theta$，其中 $0\leqslant\theta\leqslant\pi$。

(a) 画出 $C$ 的草图，并写出对称轴的方程。

(b) 求 $C$ 的一个笛卡尔方程。

(c) 求 $C$ 所围区域的总面积。

(d) 求 $C$ 上的点到极点的最大距离。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

将极径平方写成2sinθ乘cos平方，整段角区间都非负。零点把曲线分成第一、第二象限的两个环，关于纵轴对称。

Write

Here $0\leqslant\theta\leqslant\pi$. The radius is zero at $\theta=0,\frac\pi2,\pi$, and positive between these angles. Sketch one loop in the first quadrant for $0<\theta<\frac\pi2$ and its mirror image in the second quadrant for $\frac\pi2<\theta<\pi$, meeting at the pole. At the pole, each loop has one tangent along the initial line and one along the vertical axis.

Since replacing $\theta$ by $\pi-\theta$ leaves $r^2$ unchanged, the line of symmetry is

先用倍角式，再乘r三次方，右边恰好化成两个横坐标与一个纵坐标的乘积。保留二分之五次幂形式，不平方增加下半平面的额外分支。

Using $r^2=2\sin\theta\cos^2\theta$, multiply by $r^3$:

Since $r=\sqrt{x^2+y^2}$, a Cartesian equation is

This equation retains the upper-half-plane condition of the polar curve; it also includes the pole.

整个角区间恰好扫过两个环各一次，可直接积分；把正弦倍角展开后，余弦立方的导数立即出现。

The total area is

对极径平方求导，比直接对根式求导简洁。先检查三个极径为零的角，再找两个环内部的最大点。

Since $r\geqslant0$, maximising $r$ is equivalent to maximising $r^2=2\sin\theta\cos^2\theta$.

Differentiating,

The value at $\theta=0,\frac\pi2,\pi$ is $r=0$. The remaining stationary candidates satisfy

with one angle in each loop. At either point,

These positive values exceed those at all endpoints, so the greatest distance is