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CIE 9709 2025 June Paper 11 Q4

A Level / CIE / P1

CIE 9709 2025 June Paper 11 Paper · Question 4

题目

Problem

The diagram shows the curve with equation y=5x32−20xy = 5x^{\frac{3}{2}} - 20x and the line with equation y=x−16y = x - 16. The xx-coordinates of the points of intersection of the curve and line are 1 and 16.

Find the area of the shaded region between the curve and the line.

[5]
题目中文翻译

图中显示了方程为 y=5x32−20xy = 5x^{\frac{3}{2}} - 20x 的曲线和方程为 y=x−16y = x - 16 的直线。曲线与直线交点的 xx 坐标为 1 和 16。

求曲线与直线之间阴影区域的面积。

[5]

解答

解法一

思路

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阴影位于直线下方、曲线上方。按上减下写被积函数，再用两个交点作为积分限。

答题过程

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Between the intersections, the line lies above the curve. The area is

 ∫116((x−16)−(5x3/2−20x)) dx= [212x2−16x−2x5/2]116= 384−(−152)=7832.\begin{align*} &\,\int_1^{16}\bigl((x-16)-(5x^{3/2}-20x)\bigr)\,\mathrm{d}x\\ =&\,\Big[\frac{21}{2}x^2-16x-2x^{5/2}\Big]_1^{16}\\ =&\,384-\Big(-\frac{15}2\Big)=\frac{783}{2}. \end{align*}

解法二

思路

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两个图形在该区间内都位于 xx 轴下方。先求曲线与 xx 轴之间的面积，再减去直线与 xx 轴围成的三角形。

答题过程

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The curve is nonpositive on [1,16][1,16] because 5x3/2−20x=5x(x−4)5x^{3/2}-20x=5x(\sqrt x-4). Its area up to the xx-axis is

−[2x5/2−10x2]116=−((−512)−(−8))=504.-\Big[2x^{5/2}-10x^2\Big]_1^{16} =-\bigl((-512)-(-8)\bigr)=504.

The area between the line and the axis is 12(15)(15)=2252\frac12(15)(15)=\frac{225}2. Hence the shaded area is 504−2252=7832504-\frac{225}2=\frac{783}2.