题目

Problem

The diagram shows part of the curve with equation $y = \frac{1}{(5x - 4)^{\frac13}}$ and the lines $x = 2.4$ and $y = 1$ . The curve intersects the line $y = 1$ at the point $(1, 1)$ .

Find the exact volume of the solid generated when the shaded region is rotated through $360^\circ$ about the $x$ -axis.

[6]

题目中文翻译

图中显示曲线 $y = \frac{1}{(5x - 4)^{\frac13}}$ 的一部分，以及直线 $x = 2.4$ 和 $y = 1$ 。该曲线与直线 $y = 1$ 相交于点 $(1, 1)$ 。

当阴影区域绕 $x$ 轴旋转 $360^\circ$ 时，求生成立体的精确体积。

[6]

解答

解法一

思路

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把旋转体看成半径为 $1$ 的圆柱挖去曲线下方旋转形成的内层。两个体积使用同一段横坐标范围。

答题过程

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The outer cylinder has length $2.4-1=7/5$ and radius $1$ , so its volume is $7\pi/5$ . The inner volume is

\begin{align*} &\,\pi\int_1^{12/5}(5x-4)^{-2/3}\,\mathrm{d}x\\ =&\,\frac{3\pi}{5}\Big[(5x-4)^{1/3}\Big]_1^{12/5}\\ =&\,\frac{3\pi}{5}(2-1)=\frac{3\pi}{5}. \end{align*}

Therefore the required volume is $7\pi/5-3\pi/5=4\pi/5$ .

CIE 9709 2024 June Paper 11 Q9

题目

解答

解法一

思路

答题过程