NCERT Mathematics, Class 12, textbook pages 292–299 (Part II)

How to help your child with Class 12 Maths Chapter 8, Application of Integrals, at home

Areas under curves, explained for parents: a 60-minute study session, a quick challenge, a 5-day plan and the mistakes to watch for.

area between y = x and y = x²

NCERT Class 12 Maths Chapter 8 Application of Integrals: what it teaches

Your teen uses definite integrals to find areas under curves and between a curve and lines, with circles and parabolas.

Why it matters later. Area applications connect to probability and physics.

See it: this chapter in pictures

area under y = x², 0 to 3= [x³/3]₀³= 9
Integrate between limits.
x² + y² = 164 × ∫₀⁴ √(16 − x²) dx= 16π
Use symmetry.
y = x and y = x² meet at 0, 1∫₀¹ (x − x²) dx= 1/6
Top minus bottom.

You need

  • graph paper
  • a formula sheet

Tonight’s 60 minutes

  1. Area under y = x² from 0 to 3 (9).
  2. Area of a circle x² + y² = 16 by integration (16π).
  3. Sketch before integrating to choose limits.
  4. Area between y = x and y = x² (1/6).

Words to use

  • area under a curve
  • limits
  • definite integral
  • bounded region

Watch for

  • Negative areas from curves below the x-axis. Take absolute values for areas below the axis.
  • Wrong limits. Find intersection points first.

What it sounds like at home

A short example of the conversation. Your words will differ; the questions are what matter.

YouArea under y = x² from 0 to 3?

Your child[x³/3] = 9.

A game for this week: Count vs integrate

  1. Draw y = x² on graph paper.
  2. Count squares for area 0–3.
  3. Integrate.
  4. Compare.

A 5-day plan, 60 minutes a day

  1. Day 1Explain-back.
  2. Day 2Worked examples.
  3. Day 3Practice set.
  4. Day 4Real-life problem.
  5. Day 5Mixed quiz.

Everyday moments to practise

  • Land area.
  • Engineering load.
  • Economics surplus.
  • Physics work.

Quick check

Ask these at the end of the week. Tap to see the answer.

Area under y = x² from 0 to 3?

9.

Area enclosed by x² + y² = 16?

16π.

Area between y = x and y = x²?

1/6.

Ready for the next chapter when your child…

  • Sets up area integrals.
  • Finds intersection limits.
  • Computes areas correctly.

Activities, wording and illustrations are Mathify Kids’ own, following the topics of NCERT’s Mathematics (Class 12). See the full Class 12 parent guide for every chapter.

Questions parents ask

How do I find the area between two curves?

Integrate the top minus the bottom between intersection points.

Why sketch first?

To see regions and limits.

What if part is below the axis?

Treat that part’s integral as positive area.