Learning objectives
- 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 / Two-unit revision pilot
Chapter 3: Integral calculus
Chapter 3: Integral calculus: structured theory, worked examples, answered practice, and a mastery checklist for Panhellenic Exams.
CHAPTER PLAN
Learn, practise, and check your progress
Follow the steps in order or jump directly to the part you need.
—/5
completed
Estimated active study time
181 minutes
1. Understand
Objectives, key ideas, and structured theory.
Open →2. Follow the method
Worked examples that explain every step.
Open →3. Practise
Graded tasks with hints and answer guidance.
Open →4. Consolidate
Assignments from core work to challenge.
Open →5. Check
A timed, marked test with solution guidance.
Open →
Progress is stored only in this browser and does not require AI credits.
Key ideas and checks
- 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).
Unit
Core theory
The essential chapter ideas in a clear sequence before practice.
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.
Understanding the concept
An antiderivative of f is a function F with F′=f on the interval considered. All antiderivatives on an interval differ by a constant. An initial condition selects that constant; it does not change the differentiation rule.
Method and conditions
For continuous f on [a,b], calculate the definite integral as F(b)−F(a). Keep brackets around the lower-end substitution. Reversing the bounds reverses the integral's sign.
Checking and coverage boundaries
Geometric area is non-negative. Find roots or intersections first, split where the ordering changes, and integrate upper minus lower. Positive and negative contributions can cancel in a signed integral, but not in geometric area.
Original pilot material — human educator review pending
Written specifically for YourFavTeacher with AI assistance, without copying past-paper prompts. Technical and numerical checks do not replace review by a human educator. It is not an official paper or endorsed by an examination body.
Mathematics
Worked examples
Follow the method step by step and check why every step is valid.
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.
Worked example 3
f(x)=−6 on [0,5]. Is its area equal to the integral? Calculate both.
- 1F(x)=−6x; F(5)−F(0)=−6·5−0=−30
- 2The graph lies below the x-axis.
- 3E=|−30|=30
No: integral −30, area 30.
Chapter 3: Integral calculus
Practice with answers
Eight graded tasks from core fluency to exam-style application. Work independently before opening a hint or answer.
Core fluency5 minutes
Find all antiderivatives of f(x)=6x+2.
Hint
Integrate each term and add an arbitrary constant.
Answer guide
F(x)=3x²+2x+C, C∈ℝ.
Core fluency6 minutes
Evaluate ∫₀^2 6x dx.
Hint
Use antiderivative 3x² and subtract its lower-bound value from its upper-bound value.
Answer guide
3·2² − 0 = 12
Application7 minutes
Evaluate ∫_3^5 3 dx.
Hint
For a constant function, multiply the constant by the interval length.
Answer guide
3·(5−3)=6
Application8 minutes
Find F when F'(x)=6x+2 and F(0)=5.
Hint
Find all antiderivatives, then use the initial condition to determine the constant.
Answer guide
F(x)=3x²+2x+5.
Reasoning9 minutes
For f(x)=−3 on [0, 2], find the definite integral and the area to the x-axis.
Hint
The integral is signed; geometric area is non-negative.
Answer guide
Integral = −6. Area = 6 square units.
Reasoning10 minutes
For f(x)=x on [−2, 2], find the integral and the total area to the x-axis.
Hint
Split at 0 where the sign changes.
Answer guide
Integral = 0. Area = 4 square units.
Exam style11 minutes
Find the area between f(x)=3x and g(x)=2x for 0≤x≤2.
Hint
f is above g. Integrate f−g=x.
Answer guide
∫₀^2 x dx = 4/2
Exam style12 minutes
If G(x)=∫₀^x (6t+2)dt, find G'(x).
Hint
The integrand is continuous: apply the fundamental theorem of calculus.
Answer guide
G'(x)=6x+2.
Common mistakes
- Omitting C from an indefinite integral.
- Subtracting upper-bound value from lower-bound value.
- Treating a signed integral as always equal to area.
- Omitting +C when all antiderivatives are requested.
- Subtracting only one term of F(a).
- Reporting a negative geometric area.
Mastery check
- I can find and check an antiderivative.
- I can evaluate a definite integral using its bounds.
- I can distinguish an integral from geometric area.
Chapter 3: Integral calculus
Chapter homework
Six distinct assignments from core fluency to challenge, each with an estimated time, hint, and answer guide.
Core8 minutes
Find all antiderivatives of f(x)=8x+3.
Hint
Integrate each term and add an arbitrary constant.
Answer guide
F(x)=4x²+3x+C, C∈ℝ.
Core8 minutes
Evaluate ∫₀^3 8x dx.
Hint
Use antiderivative 4x² and subtract its lower-bound value from its upper-bound value.
Answer guide
4·3² − 0 = 36
Stretch8 minutes
Evaluate ∫_4^7 3 dx.
Hint
For a constant function, multiply the constant by the interval length.
Answer guide
3·(7−4)=9
Stretch8 minutes
Find F when F'(x)=8x+3 and F(0)=5.
Hint
Find all antiderivatives, then use the initial condition to determine the constant.
Answer guide
F(x)=4x²+3x+5.
Challenge8 minutes
For f(x)=−4 on [0, 3], find the definite integral and the area to the x-axis.
Hint
The integral is signed; geometric area is non-negative.
Answer guide
Integral = −12. Area = 12 square units.
Challenge8 minutes
For f(x)=x on [−3, 3], find the integral and the total area to the x-axis.
Hint
Split at 0 where the sign changes.
Answer guide
Integral = 0. Area = 9 square units.
50 minutes / 40 marks
Full chapter test
A timed, full-mark self-assessment with model-answer guidance.
Show your working and justify each theorem's assumptions. Check the answers after finishing.
1. Find all antiderivatives of f(x)=10x+4.
4 marksAnswer guide
F(x)=5x²+4x+C, C∈ℝ.
2. Evaluate ∫₀^4 10x dx.
4 marksAnswer guide
5·4² − 0 = 80
3. Evaluate ∫_5^9 3 dx.
4 marksAnswer guide
3·(9−5)=12
4. Find F when F'(x)=10x+4 and F(0)=5.
4 marksAnswer guide
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 marksAnswer guide
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 marksAnswer guide
Integral = 0. Area = 16 square units.
7. Find the area between f(x)=5x and g(x)=4x for 0≤x≤4.
4 marksAnswer guide
∫₀^4 x dx = 16/2
8. If G(x)=∫₀^x (10t+4)dt, find G'(x).
4 marksAnswer guide
G'(x)=10x+4.
9. If ∫₀^4 f(x)dx=5, what is ∫_4^0 f(x)dx?
4 marksAnswer guide
−5
10. Find the area under f(x)=x(4−x) on [0, 4].
4 marksAnswer guide
Area = [4x²/2−x³/3]₀^4 = 64/6 square units.
Unit
Official sources and verification
Curriculum reference sources. Always confirm the teaching sequence with the school and tutor.
Mathematics
Panhellenic strategy: theory, methodology, timing, grading, and continuous correction.
What this chapter covers
The structure follows the official textbook layout and is used to organise study.
Antiderivatives
Definite integrals
Geometric area and sign
Where to focus
The areas that usually create mistakes or need extra revision.
I can find and check an antiderivative.
I can evaluate a definite integral using its bounds.
I can distinguish an integral from geometric area.
Sources, daily material, and resources
Where to start: textbook, daily material, PDFs, videos, and worked examples.
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.
Practice by subtopic
Targeted practice before full tests so coverage is clear.
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 and progress checks
How to measure progress in this chapter and when it enters a cumulative 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.
Next step
What to do after finishing the chapter and how it connects to the next unit.
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
Note: for the official examinable syllabus of each school year, always confirm with the school, tutor, and current Ministry/IEP announcements.