Show all supporting work on scratch paper. On the AP Exam, Section I is split into a no-calculator and a calculator-allowed part, each question below is labeled accordingly.请在草稿纸上写出所有支持性步骤。AP 考试第一部分分为禁止使用计算器和允许使用计算器两部分，以下每题均已标注。
Q1EASY6.2 Riemann Sums6.2 黎曼和No Calculator
The interval $[0,4]$ is divided into four subintervals of equal length. Which expression gives the right Riemann sum approximation for $\displaystyle\int_{0}^{4}f(x)\,dx$?区间 $[0,4]$ 被等分为四个子区间。下列哪个表达式给出了 $\displaystyle\int_{0}^{4}f(x)\,dx$ 的右黎曼和近似值？
If $\displaystyle\int_{0}^{5}f(x)\,dx=12$ and $\displaystyle\int_{0}^{2}f(x)\,dx=3$, then $\displaystyle\int_{2}^{5}f(x)\,dx=$若 $\displaystyle\int_{0}^{5}f(x)\,dx=12$，$\displaystyle\int_{0}^{2}f(x)\,dx=3$，则 $\displaystyle\int_{2}^{5}f(x)\,dx=$
Q8MEDIUM6.6 Geometry of Integral6.6 积分的几何意义No Calculator
The graph of $f$ consists of two line segments (forming a triangle above the $x$-axis with vertices $(-2,0)$, $(0,2)$, $(2,0)$) and a semicircle of radius $2$ below the $x$-axis on $[2,6]$. Find $\displaystyle\int_{-2}^{6}f(x)\,dx$.$f$ 的图像由两条线段（在 $x$ 轴上方构成顶点为 $(-2,0)$、$(0,2)$、$(2,0)$ 的三角形）和在 $[2,6]$ 上位于 $x$ 轴下方半径为 $2$ 的半圆组成。求 $\displaystyle\int_{-2}^{6}f(x)\,dx$。
Q10MEDIUM6.3 Riemann to Integral6.3 黎曼和化定积分No Calculator
Which definite integral equals $\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\!\left(1+\dfrac{2i}{n}\right)^{2}\!\cdot\dfrac{2}{n}$?下列哪个定积分等于 $\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\!\left(1+\dfrac{2i}{n}\right)^{2}\!\cdot\dfrac{2}{n}$？
$f$ is positive, increasing, and concave down on $[a,b]$. Using only the fact that $f$ is increasing, which of the following is guaranteed to underestimate $\displaystyle\int_{a}^{b}f(x)\,dx$?$f$ 在 $[a,b]$ 上为正、单调递增且凹（下凸）。仅利用 $f$ 单调递增这一性质，下列哪项一定低估 $\displaystyle\int_{a}^{b}f(x)\,dx$？
Selected values of the differentiable function $f$ are given. Let $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$.可微函数 $f$ 的部分函数值如下表。设 $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$。
$x$
$0$
$2$
$4$
$6$
$8$
$f(x)$
$5$
$3$
$-1$
$-4$
$-2$
Using a midpoint Riemann sum with two subintervals of equal length, the approximation for $g(8)$ is用两个等长子区间的中点黎曼和近似 $g(8)$ 的值为
Q14MEDIUM6.5 Behavior of $g$6.5 累积函数 $g$ 的行为No Calculator
Let $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$, where $f$ is continuous and is positive on $(0,3)$, zero at $x=3$, and negative on $(3,5)$. At what value of $x$ does $g$ attain its maximum on $[0,5]$?设 $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$，其中 $f$ 连续，在 $(0,3)$ 上为正，在 $x=3$ 处为零，在 $(3,5)$ 上为负。$g$ 在 $[0,5]$ 上于哪个 $x$ 值处取得最大值？
Water flows into a tank at a rate $r(t)$ gallons per minute, where $t$ is in minutes. The most appropriate units of $\displaystyle\int_{0}^{10}r(t)\,dt$ are水以 $r(t)$ 加仑/分钟的速率流入水箱，其中 $t$ 以分钟计。$\displaystyle\int_{0}^{10}r(t)\,dt$ 最合适的单位是
Free-response answers must include all setup. When using a calculator, present the integral expression with limits and a differential before evaluating. When without a calculator, show antiderivatives, the constant of integration on indefinite integrals, and careful use of substitution.自由解答必须包含所有推导过程。使用计算器时，在求值前需写出含积分限和微分的积分式。不使用计算器时，需展示原函数、不定积分中的积分常数，以及换元步骤。
The rate at which water enters a reservoir is modeled by the differentiable function $R(t)$, where $R$ is in thousands of gallons per hour and $t$ is in hours since midnight. Selected values of $R$:水流入水库的速率由可微函数 $R(t)$ 建模，其中 $R$ 以千加仑/小时为单位，$t$ 为午夜后的小时数。$R$ 的部分值如下：
$t$ (hr)
$0$
$3$
$6$
$9$
$12$
$R(t)$
$5.2$
$6.8$
$8.1$
$7.4$
$4.5$
(a)Use a left Riemann sum with the four subintervals shown to approximate $\displaystyle\int_{0}^{12}R(t)\,dt$. Using correct units, explain the meaning of this integral in context.用表中四个子区间的左黎曼和近似 $\displaystyle\int_{0}^{12}R(t)\,dt$。用正确单位说明该积分在情境中的含义。
(b)Use a trapezoidal sum with the four subintervals shown to approximate $\displaystyle\int_{0}^{12}R(t)\,dt$.用表中四个子区间的梯形和近似 $\displaystyle\int_{0}^{12}R(t)\,dt$。
(c)Is the left Riemann sum in part (a) an over- or under-estimate? Explain using the behavior of $R$ shown in the table, or state what additional information would be needed to decide.第 (a) 部分的左黎曼和是高估还是低估？利用表中 $R$ 的变化趋势说明，或说明需要哪些额外信息才能判断。
(d)The reservoir holds $W(t)$ thousand gallons at time $t$. Water leaves the reservoir at a constant rate of $4$ thousand gallons per hour during the $12$-hour period. Write, but do not evaluate, an expression involving an integral for $W(12)$ given $W(0)=80$.水库在 $t$ 时刻储水 $W(t)$ 千加仑。在 12 小时内水以每小时 $4$ 千加仑的恒定速率流出。已知 $W(0)=80$，写出（但不求值）含积分的 $W(12)$ 表达式。
The graph of the continuous function $f$ on $[0,8]$ consists of three line segments and a quarter circle of radius $2$, as shown. Let $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$.连续函数 $f$ 在 $[0,8]$ 上的图像由三条线段和一个半径为 $2$ 的四分之一圆弧组成，如图所示。设 $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$。
(a)Find $g(2)$, $g(4)$, and $g(8)$.求 $g(2)$、$g(4)$ 和 $g(8)$。
(b)On what interval(s) is $g$ increasing? Justify.$g$ 在哪些区间上单调递增？请说明理由。
(c)At what value of $x$ on $[0,8]$ does $g$ attain its absolute maximum? Justify.$g$ 在 $[0,8]$ 上于哪个 $x$ 值处取得绝对最大值？请说明理由。
(d)Find the $x$-coordinate of each point of inflection of $g$ on $(0,8)$. Justify.求 $g$ 在 $(0,8)$ 上每个拐点的 $x$ 坐标。请说明理由。
Evaluate each of the following integrals, showing all algebraic steps. For $u$-substitution problems, clearly state your choice of $u$ and $du$.求以下各积分，展示所有代数步骤。换元法题目须明确写出 $u$ 和 $du$ 的选取。
Let $f$ be a continuous function on $[-2,8]$, and define $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$. It is known that $\displaystyle\int_{0}^{3}f(t)\,dt=6$, $\displaystyle\int_{3}^{5}f(t)\,dt=-2$, and $\displaystyle\int_{5}^{8}f(t)\,dt=4$.设 $f$ 是 $[-2,8]$ 上的连续函数，定义 $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$。已知 $\displaystyle\int_{0}^{3}f(t)\,dt=6$，$\displaystyle\int_{3}^{5}f(t)\,dt=-2$，$\displaystyle\int_{5}^{8}f(t)\,dt=4$。
(a)Find $g(3)$, $g(5)$, and $g(8)$.求 $g(3)$、$g(5)$ 和 $g(8)$。
(b)Find $\displaystyle\int_{-1}^{8}\bigl[\,2f(t)+1\,\bigr]\,dt$, given that $\displaystyle\int_{-1}^{0}f(t)\,dt = 1$.已知 $\displaystyle\int_{-1}^{0}f(t)\,dt = 1$，求 $\displaystyle\int_{-1}^{8}\bigl[\,2f(t)+1\,\bigr]\,dt$。
(c)Suppose $f$ is differentiable on $[0,8]$, with $f(0)=2$ and $f(8)=-3$. Let $h(x)=x\cdot f(x)$. Find $\displaystyle\int_{0}^{8}\bigl[\,f(x)+x\,f'(x)\,\bigr]\,dx$.设 $f$ 在 $[0,8]$ 上可微，$f(0)=2$，$f(8)=-3$。令 $h(x)=x\cdot f(x)$，求 $\displaystyle\int_{0}^{8}\bigl[\,f(x)+x\,f'(x)\,\bigr]\,dx$。
(d)Let $H(x)=\displaystyle\int_{0}^{x^{2}}f(t)\,dt$. Express $H'(x)$ in terms of $f$ and $x$.设 $H(x)=\displaystyle\int_{0}^{x^{2}}f(t)\,dt$，用 $f$ 和 $x$ 表示 $H'(x)$。
PART III (BC) EXTENSIONS（BC）扩展内容Topics 6.11, 6.12, 6.13 - BC ONLY考点 6.11、6.12、6.13，仅限 BC
BC-Only PracticeBC 专属练习
BC ONLY.The following items cover Integration by Parts (Topic 6.11), Linear Partial Fractions (Topic 6.12), and Improper Integrals (Topic 6.13). AB students may skip this section. BC students should be fluent with: (i) IBP $\displaystyle\int u\,dv = uv-\int v\,du$, pick $u$ via LIATE; (ii) decomposing proper rationals with distinct linear factors as $\dfrac{A}{x-r_1}+\dfrac{B}{x-r_2}+\cdots$; (iii) evaluating improper integrals as limits, including those with interior discontinuities (split first).以下题目涵盖分部积分法（考点 6.11）、线性部分分式（考点 6.12）和反常积分（考点 6.13）。AB 学生可跳过本节。BC 学生应熟练掌握：(i) 分部积分公式 $\displaystyle\int u\,dv = uv-\int v\,du$，按 LIATE 顺序选取 $u$；(ii) 将具有不同线性因子的真分式分解为 $\dfrac{A}{x-r_1}+\dfrac{B}{x-r_2}+\cdots$；(iii) 将反常积分化为极限求值，包括含内部间断点的情形（先拆分）。
(a)Decompose $\dfrac{1}{x^{2}-x}$ into partial fractions. Show the algebraic system you solve.将 $\dfrac{1}{x^{2}-x}$ 分解为部分分式，写出求解的代数方程组。
(b)Set up the integral as a limit, evaluate the antiderivative, and determine whether the integral converges. If it converges, give the exact value.将积分写成极限形式，求原函数，并判断积分是否收敛。若收敛，给出精确值。
(c)Now consider $\displaystyle\int_{0}^{2}\dfrac{1}{(x-1)^{1/3}}\,dx$, which is improper at the interior point $x=1$. Split the integral, evaluate each piece as a one-sided limit, and find the total value (or state divergence). Justify each step.现考虑 $\displaystyle\int_{0}^{2}\dfrac{1}{(x-1)^{1/3}}\,dx$，该积分在内部点 $x=1$ 处为反常积分。将积分拆分，逐段用单侧极限求值，给出总值（或说明发散）。逐步说明理由。