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Schooling · Class 12 · Mathematics · Application of Integrals · Chapter on a page

CBSE Class 12 · Mathematics · NCERT Mathematics Part-II

Application of Integrals: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

This note's current text has no applied fact-check on record.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Application of Integrals: Key ideas, Formulas and facts, Watch out forKey ideasThe definite integral ∫[a to b] f(x)…To find the actual (unsigned) area…Area between two curves y = f(x) and…When curves cross inside the…Sometimes it is simpler to integrate…Sketching the region first prevents…Formulas and factsSituationArea when curve lies below the x-axisArea between y = f(x)Area using horizontal stripsStandard results worth recalling:Area of a circle x² + y² = r² is πr²…Area of an ellipse x²/a² + y²/b² = 1…Watch out forForgetting to take the absolute…Subtracting in the wrong order…Using wrong limits because the…Confusing the roles of x and y when…Applying the area formula for a full…Application ofIntegrals
The mind map as a list
  • Key ideas
    • The definite integral ∫[a to b] f(x)…
    • To find the actual (unsigned) area…
    • Area between two curves y = f(x) and…
    • When curves cross inside the…
    • Sometimes it is simpler to integrate…
    • Sketching the region first prevents…
  • Formulas and facts
    • Situation
    • Area when curve lies below the x-axis
    • Area between y = f(x)
    • Area using horizontal strips
    • Standard results worth recalling:
    • Area of a circle x² + y² = r² is πr²…
    • Area of an ellipse x²/a² + y²/b² = 1…
  • Watch out for
    • Forgetting to take the absolute…
    • Subtracting in the wrong order…
    • Using wrong limits because the…
    • Confusing the roles of x and y when…
    • Applying the area formula for a full…

What this chapter is about

This chapter shows how definite integrals move beyond pure calculation to solve a geometric problem: finding the area of a region bounded by curves. In Class 11 and earlier in Class 12, you learned to evaluate integrals; now you apply that skill to measure plane areas that simple length × breadth cannot handle.

Key ideas

  • The definite integral ∫[a to b] f(x) dx represents the net signed area: positive where f(x) ≥ 0, negative where f(x) < 0.
  • To find the actual (unsigned) area when a curve dips below the x-axis, split the interval at the zeros and take the absolute value of each part.
  • Area between two curves y = f(x) and y = g(x) from x = a to x = b, with f(x) ≥ g(x) throughout, is ∫[a to b] [f(x) − g(x)] dx.
  • When curves cross inside the interval, locate the intersection points and split the integral accordingly, always subtracting the lower curve from the upper in each sub-interval.
  • Sometimes it is simpler to integrate with respect to y: area = ∫[c to d] [right curve − left curve] dy.
  • Sketching the region first prevents sign errors and helps identify limits of integration.

Formulas and facts to remember

  • Situation: Area under y = f(x) above the x-axis, from x = a to b · Formula: A = ∫[a to b] f(x) dx · Meaning: Sum of thin vertical strips of width dx and height f(x).
  • Situation: Area when curve lies below the x-axis · Formula: A = −∫[a to b] f(x) dx or ∫[a to b] |f(x)| dx · Meaning: The integral itself is negative; take magnitude for area.
  • Situation: Area between y = f(x) (upper) and y = g(x) (lower) · Formula: A = ∫[a to b] [f(x) − g(x)] dx · Meaning: Difference of ordinates, integrated over the common x-range.
  • Situation: Area using horizontal strips · Formula: A = ∫[c to d] [x_right(y) − x_left(y)] dy · Meaning: Useful when the boundary is easier to express as x in terms of y.
  • Standard results worth recalling:
  • Area of a circle x² + y² = r² is πr² (can be derived by integrating y = √(r² − x²) from −r to r and doubling).
  • Area of an ellipse x²/a² + y²/b² = 1 is πab.

Common mistakes

  • Forgetting to take the absolute value when the curve goes below the x-axis → split at zeros and add magnitudes.
  • Subtracting in the wrong order (lower minus upper) and getting a negative area → always upper curve minus lower curve.
  • Using wrong limits because the sketch was skipped → always draw a rough diagram first.
  • Confusing the roles of x and y when curves are easier to express as x = h(y) → switch to horizontal strips and integrate with respect to y.
  • Applying the area formula for a full circle or ellipse when only a segment or sector is asked → integrate only over the required region.

Quick revision

  • Definite integral = signed area; take absolute value for actual area below the axis.
  • Area between curves = ∫ (upper − lower) dx over the interval where they enclose a region.
  • Always find intersection points first; they become limits of integration.
  • Sketch the region to decide which function is on top.
  • Use ∫ dy (horizontal strips) when boundaries are simpler in terms of y.