Sections 1 to 7: area between curves, disk and washer volumes, cylindrical shells, cross-sections, arc length, surface area第1至7节：曲线间面积、圆盘与垫圈体积、柱壳法、截面法、弧长、曲面面积CALC II
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PART I · CORE TECHNIQUESComputational fluency · 28 marks计算熟练度 · 28分
Area Between Curves and Volumes of Revolution曲线间面积与旋转体体积
Show all working. Find intersection points before setting up any integral. When revolving, sketch the radius function and confirm it is non-negative on the interval. State which method (disk, washer, or shell) you use before integrating.展示完整解题过程。在建立积分之前先求交点。旋转时，画出半径函数并确认其在区间上非负。在积分前说明所用方法（圆盘法、垫圈法或柱壳法）。
Q1MEDIUMCOREarea between curves: integrating in $x$ and in $y$曲线间面积：关于$x$与$y$的积分[6 marks]
Let $R$ be the region enclosed by $y = 4 - x^{2}$ and $y = x + 2$.设$R$为由$y = 4 - x^{2}$与$y = x + 2$围成的区域。
(a)Find the intersection points of the two curves and write a single definite integral for the area of $R$, integrating with respect to $x$.求两条曲线的交点，并写出关于$x$的面积$R$的定积分。[3]
(b)Rewrite the boundary curves as $x = g(y)$ and express the area of $R$ as a definite integral with respect to $y$. You do not need to evaluate this second integral, but it must be correct.将边界曲线改写为$x = g(y)$的形式，并将$R$的面积表示为关于$y$的定积分。无需计算该积分，但积分式必须正确。[3]
Q2MEDIUMCOREarea requiring a split: curves that cross需要分段的面积：相交的曲线[6 marks]
Consider the curves $y = \sin x$ and $y = \cos x$ on the interval $\left[0, \tfrac{3\pi}{2}\right]$.考虑区间$\left[0, \tfrac{3\pi}{2}\right]$上的曲线$y = \sin x$与$y = \cos x$。
(a)Find all values of $x$ in $\left[0, \tfrac{3\pi}{2}\right]$ where the two curves intersect, and state which curve lies above the other on each sub-interval.求$\left[0, \tfrac{3\pi}{2}\right]$上两曲线的所有交点，并说明在每个子区间上哪条曲线位于另一条曲线的上方。[2]
(b)Write the total area between the two curves as a sum of integrals and evaluate it exactly.将两曲线间的总面积写成积分之和，并精确计算其值。[4]
Q3HARDCOREwasher method: revolution about the $x$-axis and the $y$-axis垫圈法：绕$x$轴与$y$轴旋转[8 marks]
Let $R$ be the region bounded by $y = \sqrt{x}$, $y = 0$, and $x = 4$.设$R$为由$y = \sqrt{x}$、$y = 0$与$x = 4$围成的区域。
(a)Using the disk method, find the volume of the solid generated by revolving $R$ about the $x$-axis.用圆盘法求$R$绕$x$轴旋转所得旋转体的体积。[3]
(b)Using the washer method (integrating with respect to $y$), find the volume of the solid generated by revolving $R$ about the $y$-axis. Express $x$ as a function of $y$ before writing the washer radii.用垫圈法（关于$y$积分）求$R$绕$y$轴旋转所得旋转体的体积。在写出垫圈半径之前，先将$x$表示为$y$的函数。[5]
Q4HARDCOREwasher method with offset axis: revolving about $y = k$偏移轴的垫圈法：绕$y = k$旋转[8 marks]
Let $R$ be the region enclosed by $y = x^{2}$ and $y = 2x$.设$R$为由$y = x^{2}$与$y = 2x$围成的区域。
(a)Find the intersection points of $y = x^{2}$ and $y = 2x$ and sketch the region, labelling the outer and inner curves.求$y = x^{2}$与$y = 2x$的交点，画出区域草图，并标注外曲线与内曲线。[2]
(b)Set up and evaluate the volume of the solid obtained by revolving $R$ about the line $y = -1$. Write both washer radii explicitly in terms of $x$ before integrating.建立并计算$R$绕直线$y = -1$旋转所得旋转体的体积。在积分之前，用$x$显式写出两个垫圈半径。[6]
PART II · DEFINITIONS AND PROOFRigorous derivations · 26 marks严格推导 · 26分
Derivations: Shell Volume, Arc Length, and Cross-Sections推导：柱壳体积、弧长与截面法
Arguments are graded on logic, not only the final answer. Each derivation must start from a clearly stated approximation of a thin element (shell, segment, or cross-section), express its contribution in terms of a single variable, and pass to the integral by a limiting argument.解题过程按逻辑评分，不仅看最终答案。每个推导必须从明确陈述的薄元素（壳、线段或截面）近似出发，用单一变量表达其贡献，并通过极限论证过渡到积分。
Q5HARDPROOFcylindrical shells: deriving the formula and applying it柱壳法：公式推导及其应用[8 marks]
This question derives the cylindrical-shell volume formula from first principles and then applies it.本题从第一性原理推导柱壳体积公式，并加以应用。
(a)Consider a thin cylindrical shell with inner radius $r$, outer radius $r + \Delta r$, and height $h$. Show that its volume is exactly $\pi(2r + \Delta r)h\,\Delta r$, and hence that as $\Delta r \to 0$ the volume element is $2\pi r h\,dr$.考虑一个内径为$r$、外径为$r + \Delta r$、高为$h$的薄柱壳。证明其体积恰好为$\pi(2r + \Delta r)h\,\Delta r$，并由此说明当$\Delta r \to 0$时体积元为$2\pi r h\,dr$。[3]
(b)Let $R$ be the region under $y = x(2 - x)$ for $0 \le x \le 2$, above the $x$-axis. Using the shell formula derived in (a), find the volume of the solid obtained by revolving $R$ about the $y$-axis.设$R$为$x$轴上方、$0 \le x \le 2$范围内$y = x(2 - x)$下方的区域。利用(a)中推导的柱壳公式，求$R$绕$y$轴旋转所得旋转体的体积。[5]
Q6HARDPROOFarc length: deriving the integrand from the Pythagorean approximation弧长：从勾股近似推导被积函数[8 marks]
This question derives the arc-length formula from the Pythagorean approximation of a small arc element.本题从小弧元的勾股近似推导弧长公式。
(a)On a differentiable curve $y = f(x)$, consider the chord joining $(x, f(x))$ to $(x + \Delta x, f(x + \Delta x))$. Show that its length is approximately $\sqrt{1 + [f'(x)]^{2}}\,\Delta x$ for small $\Delta x$, and hence write down the arc-length integral for $a \le x \le b$.在可微曲线$y = f(x)$上，考虑连接$(x, f(x))$与$(x + \Delta x, f(x + \Delta x))$的弦。证明当$\Delta x$很小时其长度近似为$\sqrt{1 + [f'(x)]^{2}}\,\Delta x$，并由此写出$a \le x \le b$上的弧长积分。[4]
(b)Use the formula from (a) to find the exact arc length of $y = \ln(\cos x)$ from $x = 0$ to $x = \dfrac{\pi}{4}$. You will need to simplify $1 + [f'(x)]^{2}$ under the radical before integrating.利用(a)中的公式求$y = \ln(\cos x)$从$x = 0$到$x = \dfrac{\pi}{4}$的精确弧长。积分前需先化简根号下的$1 + [f'(x)]^{2}$。[4]
Q7HARDPROOFvolumes by known cross-sections已知截面的体积[10 marks]
Each part describes a solid whose base and cross-sections are given. Use the formula $V = \displaystyle\int_{a}^{b} A(x)\,dx$, where $A(x)$ is the cross-sectional area at position $x$.各小题描述了底面和截面已知的立体。使用公式$V = \displaystyle\int_{a}^{b} A(x)\,dx$，其中$A(x)$为位置$x$处的截面面积。
(a)A solid has base the region in the $xy$-plane bounded by $y = \sqrt{x}$ and $y = \tfrac{x}{2}$, for $0 \le x \le 4$. Cross sections perpendicular to the $x$-axis are squares with one side in the base. Find the volume.一个立体的底面是$xy$平面上由$y = \sqrt{x}$与$y = \tfrac{x}{2}$围成的区域（$0 \le x \le 4$）。垂直于$x$轴的截面为正方形，其中一边位于底面内。求体积。[4]
(b)A solid has base the disk $x^{2} + y^{2} \le 4$. Cross sections perpendicular to the $x$-axis are equilateral triangles with one side as a chord of the disk at position $x$. Find the volume.一个立体的底面是圆盘$x^{2} + y^{2} \le 4$。垂直于$x$轴的截面为等边三角形，其中一边是圆盘在位置$x$处的弦。求体积。[4]
(c)State in one sentence why the general cross-section formula $V = \int A(x)\,dx$ encompasses the disk and washer formulas as special cases.用一句话说明为何一般截面公式$V = \int A(x)\,dx$将圆盘公式和垫圈公式作为特殊情形包含其中。[2]
PART III · APPLICATIONS AND SYNTHESISExtended problems · 28 marks综合应用题 · 28分
Comparing Methods, Arc Length, and Revolution about Offset Axes方法比较、弧长与绕偏移轴旋转
Set up every integral before evaluating it. For method-comparison questions, write both integrals in full and verify that both give the same volume. Carry exact values throughout; simplify under radicals algebraically before invoking any trig substitution or standard form.在计算积分之前先建立积分式。对于方法比较题，完整写出两个积分并验证两者给出相同的体积。全程保留精确值；在使用三角换元或标准形式之前，先对根号内的表达式进行代数化简。
Q8HARDAPPLIEDshell vs. washer: the same solid by two methods柱壳法与垫圈法：用两种方法求同一旋转体[10 marks]
Let $R$ be the region bounded by $y = x^{3}$, $x = 0$, and $y = 8$.设$R$为由$y = x^{3}$、$x = 0$与$y = 8$围成的区域。
(a)Sketch the region $R$, labelling the three boundary pieces.画出区域$R$的草图，并标注三条边界线。[2]
(b)Find the volume of the solid obtained by revolving $R$ about the $y$-axis using the washer method (integrate with respect to $y$).用垫圈法（关于$y$积分）求$R$绕$y$轴旋转所得旋转体的体积。[4]
(c)Find the same volume using the cylindrical-shell method (integrate with respect to $x$).用柱壳法（关于$x$积分）求相同旋转体的体积。[4]
Q9HARDAPPLIEDarc length with algebraic simplification under the radical根号下代数化简的弧长计算[10 marks]
Consider the curve $y = \dfrac{x^{3}}{6} + \dfrac{1}{2x}$ on the interval $1 \le x \le 3$.考虑区间$1 \le x \le 3$上的曲线$y = \dfrac{x^{3}}{6} + \dfrac{1}{2x}$。
(a)Compute $\dfrac{dy}{dx}$ and then $1 + \left(\dfrac{dy}{dx}\right)^{2}$. Show that the expression under the radical simplifies to a perfect square.计算$\dfrac{dy}{dx}$及$1 + \left(\dfrac{dy}{dx}\right)^{2}$。证明根号内的表达式可化简为完全平方式。[4]
(b)Hence find the exact arc length of the curve from $x = 1$ to $x = 3$.由此求曲线从$x = 1$到$x = 3$的精确弧长。[3]
(c)Briefly explain why the trick of having $1 + (f')^{2}$ be a perfect square is not a coincidence in textbook problems: what structural property of $f$ guarantees this simplification?简要说明为何$1 + (f')^{2}$恰为完全平方式在教材题目中并非巧合：$f$的哪种结构性质保证了这种化简？[3]
Q10HARDAPPLIEDrevolution about an offset axis $x = k$; shell vs. washer choice绕偏移轴$x = k$旋转：柱壳法与垫圈法的选择[8 marks]
Let $R$ be the region bounded by $y = 4 - x^{2}$ and $y = 0$, for $-2 \le x \le 2$.设$R$为$-2 \le x \le 2$范围内由$y = 4 - x^{2}$与$y = 0$围成的区域。
(a)Use the cylindrical-shell method to find the volume of the solid obtained by revolving $R$ about the line $x = 3$. Identify the shell radius carefully as a function of $x$.用柱壳法求$R$绕直线$x = 3$旋转所得旋转体的体积。仔细将壳半径表示为$x$的函数。[5]
(b)Explain in two sentences why the washer method would be significantly harder to apply here: what obstacle arises when you try to integrate with respect to $y$ against the axis $x = 3$?用两句话解释为何垫圈法在此处明显更难应用：当尝试关于$y$对轴$x = 3$积分时会遇到什么困难？[3]