图中所示为方程为 $y=5x^{\frac12}$ 的曲线和方程为 $y=2x+2$ 的直线。

求由直线与曲线围成的阴影区域面积的精确值。

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。

联立后用 $u=\sqrt x$ 求交点。图中根式曲线在上，面积是纵坐标之差的积分。

The intersection equation $5\sqrt x=2x+2$ becomes $(2u-1)(u-2)=0$ for $u=\sqrt x$. Thus the endpoints are $x=1/4$ and $x=4$. The area is

也可以取水平截面。把两条边界都写成 $x$ 关于 $y$ 的式子，用右端减左端。

The root curve is $x=y^2/25$ and the line is $x=(y-2)/2$. Their intersections have $y=5/2$ and $y=10$. On this interval the line is the right boundary, so the area is

也可先求曲线下方的面积，再扣掉直线下方的梯形面积。梯形的两条平行边是交点的纵坐标。

The intersections are $(1/4,5/2)$ and $(4,10)$. The area under the root curve between them is

The area under the straight line is a trapezium of area

Hence the shaded area is $105/4-375/16=45/16$.