学习目标
- I can find and check an antiderivative.
- I can evaluate a definite integral using its bounds.
- I can distinguish an integral from geometric area.
Panhellenic Exams / Mathematics / Curriculum
Κεφάλαιο 3: Ολοκληρωτικός Λογισμός
Κεφάλαιο 3: Ολοκληρωτικός Λογισμός: structured theory, worked examples, answered practice, and a mastery checklist for Panhellenic Exams.
章节计划
学习、练习并检查进度
按顺序完成步骤，或直接跳到你需要的部分。
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预计主动学习时间
181 分钟
1. 理解
学习目标、核心概念与结构化理论。
打开 →2. 掌握方法
逐步说明的完整例题。
打开 →3. 练习
含提示与答案的分级练习。
打开 →4. 巩固
从基础到挑战的课后任务。
打开 →5. 检查
含计时、分值与解答的章节测试。
打开 →
进度只保存在当前浏览器中，不需要 AI credits。
核心概念与检查
- An antiderivative F of f satisfies F′=f on the interval.
- Antiderivatives on an interval differ by a constant C.
- For f continuous on [a,b], ∫a^b f(x)dx=F(b)−F(a).
单元
核心理论
练习前按清晰顺序掌握本章核心概念。
Indefinite integration
Integrate term by term: ∫xⁿdx=xⁿ⁺¹/(n+1)+C for n≠−1 on a suitable interval. Check by differentiating. An additional condition is needed to determine C.
Definite integration
Find an antiderivative and evaluate upper bound minus lower bound. The result is a number. Reversing the bounds changes the sign; splitting the interval gives a sum of integrals.
Area and sign
An integral is signed. For geometric area, split where the sign changes and add non-negative areas. Between two curves, integrate the upper function minus the lower on each interval.
Mathematics
例题讲解
逐步查看方法，并理解每一步为何成立。
Evaluate a definite integral
Evaluate ∫₀² (2x+1)dx.
- 1An antiderivative is F(x)=x²+x because F′(x)=2x+1.
- 2F(2)−F(0)=(4+2)−0=6.
6
Area does not cancel
For f(x)=x on [−2,2], find the integral and the area to the x-axis.
- 1∫₋₂² x dx = [x²/2]₋₂² = 2−2=0.
- 2f is negative on [−2,0] and positive on [0,2]. Each triangular region has area 2.
- 3Add the non-negative areas: 2+2=4.
Integral 0; area 4 square units.
Κεφάλαιο 3: Ολοκληρωτικός Λογισμός
含答案练习
八道分层练习，从基础熟练到考试应用。请先独立完成，再查看提示或答案。
基础熟练5 分钟
Find all antiderivatives of f(x)=6x+2.
提示
Integrate each term and add an arbitrary constant.
答案指南
F(x)=3x²+2x+C, C∈ℝ.
基础熟练6 分钟
Evaluate ∫₀^2 6x dx.
提示
Use antiderivative 3x² and subtract its lower-bound value from its upper-bound value.
答案指南
3·2² − 0 = 12
应用7 分钟
Evaluate ∫_3^5 3 dx.
提示
For a constant function, multiply the constant by the interval length.
答案指南
3·(5−3)=6
应用8 分钟
Find F when F'(x)=6x+2 and F(0)=5.
提示
Find all antiderivatives, then use the initial condition to determine the constant.
答案指南
F(x)=3x²+2x+5.
推理9 分钟
For f(x)=−3 on [0, 2], find the definite integral and the area to the x-axis.
提示
The integral is signed; geometric area is non-negative.
答案指南
Integral = −6. Area = 6 square units.
推理10 分钟
For f(x)=x on [−2, 2], find the integral and the total area to the x-axis.
提示
Split at 0 where the sign changes.
答案指南
Integral = 0. Area = 4 square units.
考试题型11 分钟
Find the area between f(x)=3x and g(x)=2x for 0≤x≤2.
提示
f is above g. Integrate f−g=x.
答案指南
∫₀^2 x dx = 4/2
考试题型12 分钟
If G(x)=∫₀^x (6t+2)dt, find G'(x).
提示
The integrand is continuous: apply the fundamental theorem of calculus.
答案指南
G'(x)=6x+2.
常见错误
- Omitting C from an indefinite integral.
- Subtracting upper-bound value from lower-bound value.
- Treating a signed integral as always equal to area.
掌握度检查
- I can find and check an antiderivative.
- I can evaluate a definite integral using its bounds.
- I can distinguish an integral from geometric area.
Κεφάλαιο 3: Ολοκληρωτικός Λογισμός
章节作业
六道不同作业，从基础到挑战，包含预计用时、提示和答案指南。
基础8 分钟
Find all antiderivatives of f(x)=8x+3.
提示
Integrate each term and add an arbitrary constant.
答案指南
F(x)=4x²+3x+C, C∈ℝ.
基础8 分钟
Evaluate ∫₀^3 8x dx.
提示
Use antiderivative 4x² and subtract its lower-bound value from its upper-bound value.
答案指南
4·3² − 0 = 36
进阶8 分钟
Evaluate ∫_4^7 3 dx.
提示
For a constant function, multiply the constant by the interval length.
答案指南
3·(7−4)=9
进阶8 分钟
Find F when F'(x)=8x+3 and F(0)=5.
提示
Find all antiderivatives, then use the initial condition to determine the constant.
答案指南
F(x)=4x²+3x+5.
挑战8 分钟
For f(x)=−4 on [0, 3], find the definite integral and the area to the x-axis.
提示
The integral is signed; geometric area is non-negative.
答案指南
Integral = −12. Area = 12 square units.
挑战8 分钟
For f(x)=x on [−3, 3], find the integral and the total area to the x-axis.
提示
Split at 0 where the sign changes.
答案指南
Integral = 0. Area = 9 square units.
50 分钟 / 40 分
完整章节测试
带计时器、完整分值和参考答案的章节自测。
Show your working and justify each theorem's assumptions. Check the answers after finishing.
1. Find all antiderivatives of f(x)=10x+4.
4 分答案指南
F(x)=5x²+4x+C, C∈ℝ.
2. Evaluate ∫₀^4 10x dx.
4 分答案指南
5·4² − 0 = 80
3. Evaluate ∫_5^9 3 dx.
4 分答案指南
3·(9−5)=12
4. Find F when F'(x)=10x+4 and F(0)=5.
4 分答案指南
F(x)=5x²+4x+5.
5. For f(x)=−5 on [0, 4], find the definite integral and the area to the x-axis.
4 分答案指南
Integral = −20. Area = 20 square units.
6. For f(x)=x on [−4, 4], find the integral and the total area to the x-axis.
4 分答案指南
Integral = 0. Area = 16 square units.
7. Find the area between f(x)=5x and g(x)=4x for 0≤x≤4.
4 分答案指南
∫₀^4 x dx = 16/2
8. If G(x)=∫₀^x (10t+4)dt, find G'(x).
4 分答案指南
G'(x)=10x+4.
9. If ∫₀^4 f(x)dx=5, what is ∫_4^0 f(x)dx?
4 分答案指南
−5
10. Find the area under f(x)=x(4−x) on [0, 4].
4 分答案指南
Area = [4x²/2−x³/3]₀^4 = 64/6 square units.
单元
官方来源与核验
课程参考来源；教学顺序请始终与学校和老师确认。
Mathematics
Panhellenic Exams: 数学与物理的结构化路线，包含章节、练习和下一步。
本章覆盖内容
结构跟随教材，用来帮助组织学习。
Αρχική συνάρτηση στο τμήμα που προβλέπεται
Ορισμένο ολοκλήρωμα
Ιδιότητες ολοκληρώματος
Συνάρτηση ολοκλήρωμα
Θεμελιώδες Θεώρημα Ολοκληρωτικού Λογισμού
Εμβαδόν επίπεδου χωρίου
Εξαιρέσεις/περιορισμοί όπως ορίζονται στην επίσημη εξεταστέα ύλη
重点关注
通常容易出错或需要更多复习的部分。
I can find and check an antiderivative.
I can evaluate a definite integral using its bounds.
I can distinguish an integral from geometric area.
来源、日常资料与学习资源
从教材、daily material、PDFs、videos 和 worked examples 开始。
Start from the official textbook or specification referenced on the subject page.
Use the notes and examples as support, not as a replacement for the official syllabus.
Check the current syllabus version before exam preparation.
按小主题练习
进入 full tests 前先做 targeted practice，让覆盖情况更清楚。
Find all antiderivatives of f(x)=6x+2.
Evaluate ∫₀^2 6x dx.
Evaluate ∫_3^5 3 dx.
Find F when F'(x)=6x+2 and F(0)=5.
For f(x)=−3 on [0, 2], find the definite integral and the area to the x-axis.
For f(x)=x on [−2, 2], find the integral and the total area to the x-axis.
Find the area between f(x)=3x and g(x)=2x for 0≤x≤2.
If G(x)=∫₀^x (6t+2)dt, find G'(x).
Mocks 与进度检查
如何衡量本章进度，以及何时加入综合 mock。
Start with untimed practice by subtopic.
Move to a short timed checkpoint only after completing the mastery checklist.
Record each error with the correct method and revisit it after 48 hours.
下一步
完成本章后该做什么，以及它如何连接到下一单元。
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
说明：每学年的正式考试范围请始终以学校、老师和当前官方来源为准。