图中所示的是曲线 $y=3\sqrt{2x+1}$、曲线在点 $R\,(4,9)$ 处的法线，以及直线 $y=a$，其中 $a$ 为正常数。

(a) 求阴影区域的面积。

(b) 当 $y$ 轴、曲线在 $R$ 处的法线和直线 $y=a$ 围成的三角形面积等于 9(a) 中求得的面积时，求 $a$ 的精确值。

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。

阴影从 $y$ 轴延伸到过 $R$ 的竖线。积分求这段曲线下方的面积。

The shaded area is

也可取宽 $4$、高 $9$ 的长方形，用横向积分扣掉左上方的空白。

For $3\le y\le9$, the curve has $x=y^2/18-1/2$. Thus the rectangle of area $36$ contains an unshaded portion of area

The shaded area is $36-10=26$.

求出法线，再看它与 $y$ 轴及水平线 $y=a$ 围成的直角三角形。图中的水平线在 $R$ 下方。

The curve gradient is $3/\sqrt{2x+1}$, equal to $1$ at $R$. The normal is therefore $y-9=-(x-4)$, or $y=13-x$. It meets the $y$-axis at $(0,13)$ and meets $y=a$ at $(13-a,a)$. In the configuration shown, $a<9$, so both perpendicular side lengths of the triangle are $13-a$. Equating its area to $26$ gives

Thus $13-a=\sqrt{52}$ and $a=13-2\sqrt{13}$, which is positive and below $9$.