Sections 1 to 7: antiderivatives, Riemann sums, the definite integral, FTC Parts 1 and 2, substitution, average value第 1 至 7 节：不定积分、黎曼和、定积分、微积分基本定理第1和第2部分、换元法、平均值CALC I
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PART I · CORE TECHNIQUES第一部分 · 核心计算技巧Computational fluency · 28 marks计算熟练度 · 28分
Antiderivatives, Riemann Sums, and the FTC不定积分、黎曼和与微积分基本定理
Show all working. Always include $+C$ on indefinite integrals. For definite integrals, write the antiderivative in bracket notation before substituting limits. State the rule or technique you use at each step.写出完整解题过程。不定积分必须加上 $+C$。计算定积分时，先用括号记号写出原函数，再代入上下限。每一步均需注明所用规则或方法。
Q1MEDIUMCOREindefinite integrals and initial conditions不定积分与初始条件[6 marks]
Find the general antiderivative of each function. In part (c), also find the particular antiderivative satisfying the given initial condition.求每个函数的一般不定积分。在第 (c) 部分，还需求满足给定初始条件的特解。
Q2MEDIUMCOREleft, right, and midpoint Riemann sums左、右端点及中点黎曼和[6 marks]
Let $f(x) = x^{2} + 1$ on $[1, 3]$. Use a regular partition with $n = 4$ equal subintervals.设 $f(x) = x^{2} + 1$ 在 $[1, 3]$ 上。使用 $n = 4$ 个等分子区间的正则划分。
(a)State $\Delta x$ and the four partition points $x_{0}, x_{1}, x_{2}, x_{3}, x_{4}$.写出 $\Delta x$ 及四个分点 $x_{0}, x_{1}, x_{2}, x_{3}, x_{4}$。[1]
(b)Compute the left Riemann sum $L_{4}$ and the right Riemann sum $R_{4}$.计算左端点黎曼和 $L_{4}$ 与右端点黎曼和 $R_{4}$。[3]
(c)The exact value of $\displaystyle\int_{1}^{3}(x^{2}+1)\,dx$ is $\tfrac{32}{3}$. State which of $L_{4}$ and $R_{4}$ is an overestimate and which is an underestimate, and explain why in terms of the monotonicity of $f$.$\displaystyle\int_{1}^{3}(x^{2}+1)\,dx$ 的精确值为 $\tfrac{32}{3}$。指出 $L_{4}$ 和 $R_{4}$ 哪个是高估，哪个是低估，并结合 $f$ 的单调性加以解释。[2]
Q3MEDIUMCOREFTC Part 2 and $u$-substitution微积分基本定理第2部分与 $u$-换元法[8 marks]
Evaluate each integral. For (c) and (d), use substitution and either change the limits to $u$ values or return to the $x$ variable before evaluating.计算每个积分。对于 (c) 和 (d)，使用换元积分法，可将积分限换为 $u$ 的值，或在代值前将变量换回 $x$。
(a) $\displaystyle\int_{0}^{\pi} (3\cos x - 2x)\,dx$ [2]
Q4HARDCOREproperties of the definite integral, signed and total area定积分的性质、有符号面积与总面积[8 marks]
Let $f(x) = x^{2} - 4$ on $[-2, 4]$.设 $f(x) = x^{2} - 4$ 在 $[-2, 4]$ 上。
(a)Find the zeros of $f$ in $[-2, 4]$ and determine on which subintervals $f$ is positive and on which it is negative.求 $f$ 在 $[-2, 4]$ 上的零点，并判断 $f$ 在哪些子区间上为正、哪些子区间上为负。[2]
(b)Evaluate $\displaystyle\int_{-2}^{4}(x^{2}-4)\,dx$ using FTC Part 2. This is the signed area.用微积分基本定理第2部分计算 $\displaystyle\int_{-2}^{4}(x^{2}-4)\,dx$，此为有符号面积。[2]
(c)Given that $\displaystyle\int_{0}^{4}(x^{2}-4)\,dx = \frac{16}{3}$, use properties of the integral to find $\displaystyle\int_{-2}^{0}(x^{2}-4)\,dx$ without any new antidifferentiation.已知 $\displaystyle\int_{0}^{4}(x^{2}-4)\,dx = \frac{16}{3}$，利用积分的性质求 $\displaystyle\int_{-2}^{0}(x^{2}-4)\,dx$，无需重新求原函数。[2]
(d)Compute the total (unsigned) geometric area enclosed between the graph of $f$ and the $x$-axis on $[-2, 4]$. Justify how this differs from the signed integral in (b).计算 $f$ 的图像与 $x$ 轴在 $[-2, 4]$ 上所围的总（无符号）几何面积，并说明其与 (b) 中有符号积分的区别。[2]
PART II · DEFINITIONS AND PROOF第二部分 · 定义与证明Rigorous arguments · 26 marks严格论证 · 26分
Riemann-Sum Limits and the Fundamental Theorem黎曼和的极限与微积分基本定理
These items are graded on the rigour and logic of the argument, not just the final answer. In the Riemann-sum limit, carry all algebraic steps. For FTC proofs, state every hypothesis you invoke before applying a theorem.本部分按论证的严谨性和逻辑评分，不仅看最终答案。在求黎曼和极限时，需写出全部代数步骤。在证明微积分基本定理时，每次应用定理前需明确陈述所引用的假设。
Q5HARDPROOFdefinite integral as a limit of Riemann sums定积分作为黎曼和的极限[8 marks]
Use the limit-of-right-Riemann-sums definition of the definite integral throughout this question. Do not use FTC Part 2 to shortcut any computation.本题全程使用右端点黎曼和极限来定义定积分，不得用微积分基本定理第2部分简化任何计算。
(a)Express $\displaystyle\int_{0}^{3} x^{2}\,dx$ as $\displaystyle\lim_{n\to\infty} R_{n}$ where $R_n$ is the $n$-term right Riemann sum. Write $R_n$ in closed form using the identity $\displaystyle\sum_{i=1}^{n} i^{2} = \frac{n(n+1)(2n+1)}{6}$.将 $\displaystyle\int_{0}^{3} x^{2}\,dx$ 表示为 $\displaystyle\lim_{n\to\infty} R_{n}$，其中 $R_n$ 为 $n$ 项右端点黎曼和。利用恒等式 $\displaystyle\sum_{i=1}^{n} i^{2} = \frac{n(n+1)(2n+1)}{6}$ 将 $R_n$ 写成封闭形式。[4]
(b)Evaluate the limit as $n \to \infty$ to obtain the exact value of $\displaystyle\int_{0}^{3} x^{2}\,dx$.对 $n \to \infty$ 求极限，得到 $\displaystyle\int_{0}^{3} x^{2}\,dx$ 的精确值。[2]
(c)Identify what the summation identity $\displaystyle\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$ would be needed for if you instead computed $\displaystyle\int_{0}^{3} x\,dx$ by the same method, and state the answer for that integral.说明若用同样方法计算 $\displaystyle\int_{0}^{3} x\,dx$，求和恒等式 $\displaystyle\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$ 将在何处用到，并写出该积分的值。[2]
Q6HARDPROOFFundamental Theorem of Calculus Part 1: statement, proof, and chain rule微积分基本定理第1部分：陈述、证明与链式法则[10 marks]
This question develops FTC Part 1 from first principles and then extends it.本题从基本原理出发建立微积分基本定理第1部分，并将其推广。
(a)State FTC Part 1 precisely: give the hypotheses on $f$ and the conclusion about $g(x) = \displaystyle\int_{a}^{x} f(t)\,dt$.精确陈述微积分基本定理第1部分：给出对 $f$ 的假设条件及关于 $g(x) = \displaystyle\int_{a}^{x} f(t)\,dt$ 的结论。[2]
(b)Prove the conclusion of FTC Part 1. Begin by writing out $\dfrac{g(x+h)-g(x)}{h}$ as an integral, then use the Extreme Value Theorem to bound the integrand and apply the squeeze theorem as $h \to 0$.证明微积分基本定理第1部分的结论。首先将 $\dfrac{g(x+h)-g(x)}{h}$ 写成积分形式，然后利用极值定理对被积函数进行界定，最后在 $h \to 0$ 时应用夹逼定理。[5]
(c)Let $G(x) = \displaystyle\int_{1}^{x^{3}} \frac{1}{1+t^{2}}\,dt$. Use the chain rule extension of FTC Part 1 to find $G'(x)$.设 $G(x) = \displaystyle\int_{1}^{x^{3}} \frac{1}{1+t^{2}}\,dt$。利用微积分基本定理第1部分的链式法则推广求 $G'(x)$。[3]
Q7HARDPROOFFundamental Theorem of Calculus Part 2: derivation and application微积分基本定理第2部分：推导与应用[8 marks]
This question builds FTC Part 2 from Part 1 and applies it.本题从第1部分出发建立微积分基本定理第2部分，并加以应用。
(a)Let $F$ be any antiderivative of the continuous function $f$ on $[a,b]$, and let $g(x) = \displaystyle\int_{a}^{x} f(t)\,dt$. Using only FTC Part 1 and the theorem that two functions with identical derivatives on an interval differ by a constant, derive the evaluation formula $\displaystyle\int_{a}^{b} f(x)\,dx = F(b) - F(a)$.设 $F$ 是连续函数 $f$ 在 $[a,b]$ 上的任意一个原函数，$g(x) = \displaystyle\int_{a}^{x} f(t)\,dt$。仅利用微积分基本定理第1部分以及"在区间上导数相同的两个函数相差一个常数"的定理，推导求值公式 $\displaystyle\int_{a}^{b} f(x)\,dx = F(b) - F(a)$。[5]
(b)A particle moves along a straight line with velocity $v(t) = t^{2} - 3t + 2$ metres per second for $0 \le t \le 3$. Use FTC Part 2 to find the net displacement and the total distance travelled over this time interval.一质点沿直线运动，速度为 $v(t) = t^{2} - 3t + 2$ 米/秒，$0 \le t \le 3$。用微积分基本定理第2部分求该时间段内的净位移和总路程。[3]
PART III · APPLICATIONS AND SYNTHESIS第三部分 · 应用与综合Extended problems · 28 marks综合题 · 28分
Variable-Limit Integrals, Accumulation Functions, and Average Value变上限积分、累积函数与平均值
Set up each problem cleanly. Carry exact values through intermediate steps. When analysing an accumulation function, reason directly from the sign and behaviour of the integrand.清晰地建立每道题的解题框架。中间步骤保留精确值。分析累积函数时，直接从被积函数的符号和行为出发进行推断。
Q8HARDAPPLIEDFTC Part 1 with composite upper limits微积分基本定理第1部分与复合上限[8 marks]
Differentiate each expression. In each case, identify $f(t)$, the upper-limit function $u(x)$, and apply the chain rule correctly. Simplify your answer.对每个表达式求导。在每种情况下，识别 $f(t)$、上限函数 $u(x)$，并正确应用链式法则。化简最终结果。
(c)$\displaystyle H(x) = \int_{x}^{x^{2}} \cos(t^{2})\,dt$. Find $H'(x)$ by first splitting the integral at $t = 0$.$\displaystyle H(x) = \int_{x}^{x^{2}} \cos(t^{2})\,dt$。先在 $t = 0$ 处拆分积分，再求 $H'(x)$。[4]
Q9HARDAPPLIEDaccumulation function analysis: monotonicity and concavity累积函数分析：单调性与凹凸性[10 marks]
Let $\displaystyle F(x) = \int_{0}^{x} f(t)\,dt$, where $f(t) = t(t-2)(t-4)$ and the domain of $F$ is $[0, 5]$. You do not need to evaluate $F(x)$ in closed form for parts (a) through (d).设 $\displaystyle F(x) = \int_{0}^{x} f(t)\,dt$，其中 $f(t) = t(t-2)(t-4)$，$F$ 的定义域为 $[0, 5]$。第 (a) 至 (d) 部分无需将 $F(x)$ 写成封闭形式。
(a)Find all $x$ in $[0, 5]$ where $F'(x) = 0$. Classify each as a local maximum, local minimum, or neither for $F$, with a sign chart for $F'$.求 $[0, 5]$ 内所有使 $F'(x) = 0$ 的 $x$。借助 $F'$ 的符号表，将每个点分类为 $F$ 的极大值、极小值或两者均非。[3]
(b)Find all $x$ in $[0, 5]$ where $F''(x) = 0$ and determine on which subintervals $F$ is concave up and concave down.求 $[0, 5]$ 内所有使 $F''(x) = 0$ 的 $x$，并判断 $F$ 在哪些子区间上凹、在哪些子区间上凸。[3]
(c)Explain, citing FTC Part 1, why $F$ is decreasing on $(2, 4)$ even though $F(0) = 0$ and $F$ is an integral of a cubic.引用微积分基本定理第1部分，解释为何 $F$ 在 $(2, 4)$ 上单调递减，尽管 $F(0) = 0$ 且 $F$ 是一个三次函数的积分。[2]
(d)Use FTC Part 2 to find the exact value of $F(4) = \displaystyle\int_{0}^{4} t(t-2)(t-4)\,dt$.用微积分基本定理第2部分求 $F(4) = \displaystyle\int_{0}^{4} t(t-2)(t-4)\,dt$ 的精确值。[2]
Q10HARDAPPLIEDaverage value, Riemann-sum-to-integral conversion, and substitution synthesis平均值、黎曼和转化为定积分与换元法综合[10 marks]
This question connects the main ideas of the unit.本题综合本单元的核心思想。
(a)Find the average value of $f(x) = \sqrt{4 - x^{2}}$ on $[-2, 2]$. Use the geometric interpretation of the integral to evaluate $\displaystyle\int_{-2}^{2}\sqrt{4-x^{2}}\,dx$ without antidifferentiating, then apply the average value formula.求 $f(x) = \sqrt{4 - x^{2}}$ 在 $[-2, 2]$ 上的平均值。利用积分的几何意义在不求原函数的情况下计算 $\displaystyle\int_{-2}^{2}\sqrt{4-x^{2}}\,dx$，再应用平均值公式。[3]
(b)Identify the definite integral that equals $\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{n}\cdot\frac{i}{n}\cdot e^{(i/n)^{2}}$. Write the integral in the form $\displaystyle\int_{a}^{b} g(x)\,dx$ and then evaluate it using the substitution $u = x^{2}$.识别等于 $\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{n}\cdot\frac{i}{n}\cdot e^{(i/n)^{2}}$ 的定积分。将该积分写成 $\displaystyle\int_{a}^{b} g(x)\,dx$ 的形式，再用换元法 $u = x^{2}$ 求值。[4]
(c)A temperature reading $T(t) = 20 + 8\sin\!\left(\dfrac{\pi t}{12}\right)$ degrees Celsius models a day from $t = 0$ to $t = 12$ hours. Find the average temperature over this period.温度函数 $T(t) = 20 + 8\sin\!\left(\dfrac{\pi t}{12}\right)$（摄氏度）描述了从 $t = 0$ 到 $t = 12$ 小时的一天温度变化。求该时段内的平均温度。[3]