题目

Problem

The diagram shows the curves with equations $y = x^3 - 3x + 3$ and $y = 2x^3 - 4x^2 + 3$ .

(a) Find the $x$ -coordinates of the points of intersection of the curves.

[3]

(b) Find the area of the shaded region.

[4]

题目中文翻译

图中显示方程为 $y = x^3 - 3x + 3$ 和 $y = 2x^3 - 4x^2 + 3$ 的曲线。

(a) 求两条曲线交点的 $x$ 坐标。

(b) 求阴影区域的面积。

解答

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交点处两条曲线的纵坐标相等。移项后先提出公因式 $x$ ，不要除掉 $x=0$ 这个解。

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Equating the two expressions for $y$ gives

x^3-3x+3=2x^3-4x^2+3 \quad\Longrightarrow\quad x(x-1)(x-3)=0.

The intersection coordinates in the $x$ -direction are $0$ , $1$ and $3$ .

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图中只涂了 $x=1$ 到 $x=3$ 之间的区域。用上方曲线减去下方曲线，再在这两个交点之间积分。

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On $1<x<3$ , the difference $(2x^3-4x^2+3)-(x^3-3x+3)=x(x-1)(x-3)$ is negative. Thus the shaded area is

\begin{align*} &\,\int_1^3(-x^3+4x^2-3x)\,\mathrm{d}x\\ =&\,\Big[-\frac{x^4}{4}+\frac{4x^3}{3}-\frac{3x^2}{2}\Big]_1^3\\ =&\,\frac94-\Big(-\frac5{12}\Big)=\frac83. \end{align*}