题目

Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of the curve with equation

y=\frac{4}{x+2}\qquad x>-2

The region $R$ , bounded by the curve, the $y$ -axis, the $x$ -axis and the line with equation $x=8$ is shown shaded in Figure 1.

Region $R$ is rotated through 360 degrees about the $x$ -axis.

Use calculus to find the exact value of the volume of the solid generated, writing your answer in simplest form.

(5)

题目中文翻译

本题必须写出全部解题步骤。

不接受完全依赖计算器技术的解法。

图 1 给出了曲线的草图，其方程为

y=\frac{4}{x+2}\qquad x>-2

阴影部分区域 $R$ 由该曲线、 $y$ 轴、 $x$ 轴以及直线 $x=8$ 围成，如图 1 所示。

将区域 $R$ 绕 $x$ 轴旋转 360 度。

用微积分求所生成立体的体积精确值，并将答案写成最简形式。

解答

The region is rotated about the $x$ -axis, so the volume is

V=\pi\int_0^8 y^2\,\mathrm{d}x

Here

y=\frac{4}{x+2}

Therefore

\begin{aligned} V &=\pi\int_0^8 \left(\frac{4}{x+2}\right)^2\,\mathrm{d}x \\ &=16\pi\int_0^8 (x+2)^{-2}\,\mathrm{d}x \end{aligned}

Integrate:

\int (x+2)^{-2}\,\mathrm{d}x =-(x+2)^{-1} =-\frac{1}{x+2}

\begin{aligned} V &=16\pi\left[-\frac{1}{x+2}\right]_0^8 \\ &=16\pi\left(-\frac1{10}-\left(-\frac12\right)\right) \\ &=16\pi\left(\frac12-\frac1{10}\right) \\ &=16\pi\cdot\frac25 \\ &=\frac{32\pi}{5} \end{aligned}

Hence the exact volume is

\boxed{\frac{32\pi}{5}}