Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Evaluate definite integrals
Calculus
CCSS HSF.IF.B.6England: A-level Pure Mathematics
Objective and success criteria
Students will: Evaluate definite integrals.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Integrate powers of x (indefinite integrals)
Leads to: Find the area between a curve and the x-axis
Materials: Graphing tool or graph paper, Scientific calculators, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (Integrate powers of x (indefinite integrals)). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find ∫ −6x² dx. Leave out the + C. → -2*x^3Prompt: Add 1 to the power, then divide by the new power.
- Find ∫ (6x² − 6x + 9) dx. Leave out the + C. → 2x^3-3x^2+9xPrompt: Integrate each term separately.
- Ethan needs ∫ 2/x² dx. Which answer is correct? (2/x + C / 2/x³ + C / −2/x + C / 2·ln x + C) → −2/x + CPrompt: Rewrite as 2x^(−2).
Hook · 5 min
- Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
- Share the objective: Evaluate definite integrals. Read the success criteria together.
I do: model it · 10 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Evaluate ∫ from 0 to 2 of (6x² − 3) dx.
- Step 1: [2x³ − 3x] from 0 to 2
- Step 2: 10 − 0 = 10
- Answer: 10
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 10 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Evaluate ∫ from 1 to 2 of (12x² + 2) dx. → 30Prompt: Integrate, then work out F(upper) − F(lower).
- Evaluate ∫ from 0 to 2 of (x² + x) dx. Give an exact answer (a fraction is fine). → 14/3 (4 2/3)Prompt: Integrate each term.
You do: independent practice · 10 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 5 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Evaluate ∫ from 0 to 1 of (6x²) dx. → 2Prompt: Integrate, then work out F(upper) − F(lower).
- Evaluate ∫ from −2 to 0 of (3x² − 4x) dx. Give an exact answer (a fraction is fine). → 16Prompt: Integrate each term.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Integrate powers of x (indefinite integrals).
- Use concrete materials: graphing tool or graph paper.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Move on to: Find the area between a curve and the x-axis.
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.
Free lesson plan from eurekami.com/lesson-plan-builder