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CBSE Class 12 · Mathematics

Application of Integrals

Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.

This lesson follows the official chapter “Application of Integrals” in Mathematics Part I–II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 12
  • Medium level
  • 8 concepts

1Area under Simple Curves

Area under simple curves: the definite integral of y=f(x) from a to b is the signed area as taught. Under y=x from 0 to 2 is 2. Area asked as a size uses the absolute value if the lesson said so. Area is a definite-integral job, not a triangle-formula steal unless the curve is a line you already know.

Using (1/2)bh on y=x^2 is a miss.

Figure. Area under y=x² from 0 to 2 is the region between the curve, the x-axis, and the ordinate at 2. The word area sits under the curve; a rectangle would lie about the boundary.

How it works

  1. Write the curve and the x-limitsThe given.
  2. Form the definite integralThe area.
  3. Take absolute value if a size was askedHonest.

Area under y=x

Area under y=x from 0 to 2.

  • Antiderivativex^2/2
  • Evaluate4/2 - 0 = 2
  • Readarea 2

Pro tip. FTC on the curve.

2A definition is a test you can run

Area-word is a test: a definite integral under a named curve, or between two taught curves. If you only say “region”, you have a heading.

A shaded-sticker is not the test.

Figure. The definition is a test: name the curve and the ends, then the area is the number that test returns. A pretty graph without a and b has not yet been tested.

How it works

  1. Name the curve and the limitsThe object.
  2. Give the school sentenceThe test.
  3. Then the word has contentThe definition ran.

3Name the given before the unknown

The given is y=f(x) and a,b. The unknown is the area. Copy y=x from 0 to 2 before you treat it as y=x^2.

Using 0 to 1 when the figure said 0 to 2 is a silent skip.

Figure. Name the given curve and the two ends before the unknown area. For y=x², a=0, b=2 the unknown is one number, not a new curve.

How it works

  1. Copy the curve and limitsThe given.
  2. Name the integralThe unknown.
  3. Then evaluateGiven first.

4One worked case is enough at this class

One worked case is enough: area 2 under y=x from 0 to 2. Do not stack four pretty regions. The case teaches integral-as-area.

A poster of ten shadings is not more science.

Figure. One worked interval is enough at this class: area of y=x² from 0 to 2 is 8/3. A second curve is not a second idea.

How it works

  1. Take one curve and two limitsThe case.
  2. Evaluate the definite integralThe teach.
  3. Stop — one caseThis class.

5A miss: swapping the school name for the picture

A miss: swapping the school name for a picture. “Area under the curve” is the integral — a green shade is a setting.

A pretty shade without limits is a mute picture.

Figure. The graph is a picture of y. The area is the number under it on [a, b]. Calling the sketch itself “the area” swaps the school name for the picture.

How it works

  1. Keep the integral as the nameThe science.
  2. Use a shade only as a settingHonest.
  3. Refuse a swap of name for glowThe miss named.

6Check by the opposite action or the opposite test

Check by the opposite: differentiate the antiderivative to recover f; a negative signed area means the curve sat below the axis as taught.

A positive-looking shade with a below-axis curve and no sign-talk is a weak check.

Figure. Check an antiderivative by the opposite action: differentiate x³/3 and demand x², the integrand you started with.

How it works

  1. Differentiate F back to fThe check.
  2. Read a sign if the curve crossed the axisHonest.
  3. If F' is not f, redoThe opposite test.

7Keep the claim at this chapter, not the next

Keep the claim at this chapter: area under simple curves. It does not start differential equations or a volume-dump.

A DE-solve here is a chapter-swap.

Figure. This chapter’s claim is a plane region between a curve and the axis. Volume of a solid is a later claim — keep it off this page.

How it works

  1. Stay on area-as-integralThis chapter.
  2. Leave DE and later mapsThe boundary.
  3. Refuse a volume-of-revolution dump unless taughtThis class.

8The story is the hook; the test is the idea

The story is the hook; the test is the idea: a fence-under-a-graph story is a setting. The idea is still F(b)-F(a) for that f.

A story with no integral is a mute hook.

Figure. The story (a curve over an interval) is the hook. The idea is the area test that interval asks for — not a new tale.

How it works

  1. Keep the story as a hookA setting.
  2. Run F(b)-F(a)The idea.
  3. Refuse a story as the whole scienceThe test.
Area under y=x from 0 to 2 is
  1. 2
  2. 4
  3. 1

Evaluate x^2/2.

Notes

  • The official chapter title is “Application of Integrals”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • area under f from a to b = F(b)-F(a)

Recap

Hold these pegs from the official chapter “Application of Integrals”. The wording is ExamMaster’s teaching, not a textbook recap.

Area under Simple Curves
Area under simple curves: the definite integral of y=f(x) from a to b is the signed area as taught.
A definition is a test you can run
Area-word is a test: a definite integral under a named curve, or between two taught curves.
Name the given before the unknown
The given is y=f(x) and a,b. The unknown is the area. Copy y=x from 0 to 2 before you treat it as y=x^2.
One worked case is enough at this class
One worked case is enough: area 2 under y=x from 0 to 2.
A miss: swapping the school name for the picture
A miss: swapping the school name for a picture.
Check by the opposite action or the opposite test
Check by the opposite: differentiate the antiderivative to recover f; a negative signed area means the curve sat below the axis as taught.

Practise Application of Integrals

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  • A 4-question practice set that ends the chapter
  • 1 quick check with worked explanations
  • Timed mocks scored with the real marking scheme
  • Readiness tracked per topic, kept on your device
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