Math lesson plan · 12th grade (Year 13 in England) · 30 minutes
Find the area between a curve and the x-axis
Calculus
CCSS HSF.IF.B.6England: A-level Pure Mathematics
Objective and success criteria
Students will: Find the area between a curve and the x-axis.
- 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: Evaluate definite integrals
Leads to: the next skill in this area.
Materials: Graphing tool or graph paper, Scientific calculators, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 4 min
- Quick retrieval from earlier learning (Evaluate definite integrals). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Evaluate ∫ from 1 to 4 of (6x² − 2) dx. → 120Prompt: Integrate, then work out F(upper) − F(lower).
- Evaluate ∫ from 0 to 3 of (2x²) dx. Give an exact answer (a fraction is fine). → 18Prompt: Integrate each term.
- Evaluate ∫ from 1 to 2 of (6x² − 4) dx. → 10Prompt: Integrate, then work out F(upper) − F(lower).
Hook · 3 min
- Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
- Share the objective: Find the area between a curve and the x-axis. Read the success criteria together.
I do: model it · 7 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Find the area between y = x² + 2, the x-axis, x = 0 and x = 6.
- Step 1: [x³/3 + 2x] from 0 to 6
- Step 2: = 84
- Answer: 84
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 6 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Find the area between y = x² + 7, the x-axis, x = 0 and x = 6. → 114Prompt: The curve is above the axis, so area = ∫ y dx.
- The curve y = −(x + 3)(x − 1) is above the x-axis between its roots. Find the area of that region. Give an exact answer. → 32/3 (10 2/3)Prompt: Expand: y = −x² − 2x + 3.
You do: independent practice · 7 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 6 questions are on the student sheet below, with answers in the key.
Exit ticket · 3 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- The curve y = −(x + 1)(x − 5) is above the x-axis between its roots. Find the area of that region. Give an exact answer. → 36Prompt: Expand: y = −x² + 4x + 5.
- Find the area between y = x² + 9, the x-axis, x = 0 and x = 9. → 324Prompt: The curve is above the axis, so area = ∫ y dx.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Evaluate definite 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
- 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