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Mathematics · Class 12 Science

Ch 12Application of Definite Integration — Class 12 Mathematics, concept-first.

The previous chapter defined the definite integral as the limit of a sum, and proved the Fundamental Theorem of Integral Calculus: if , then . This chapter's whole purpose is to read that same number geometrically -- as an area -- and to use it to measure the plane regions trapped by curves that appear in Class 12 math…

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Key concepts

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Area under a Curve

The area under a curve , bounded by the X-axis and two vertical lines and , is found by slicing the region into a very large number of extremely thin vertical strips, each of width and height , and adding up their areas.…

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Area under the curve

The previous chapter defined the definite integral as the limit of a sum, and proved the Fundamental Theorem of Integral Calculus: if , then .

5.1.1

Area under a curve

This section states, side by side, the two basic forms the area formula can take, and then works through the chapter's first complete example.

5.1.2

Area between two curves

This is the chapter's longest section: it extends the single-curve idea of 5.1.1 to two curves at once, works out what happens when a curve dips below the X-axis or changes sign partway through an int…

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 10 questions10 questions
  1. Q1Find the area bounded by the curve $y^2 = 4ax$, X-axis and the lines $x = 0$ and $x = a$.Preview
  2. Q2Find the area of the region lying between the parabolas $y^2 = 4ax$ and $x^2 = 4ay$.Preview
  3. Q3Find the area of ellipse $\dfrac{x^2}{1} + \dfrac{y^2}{4} = 1$.Preview
  4. Q4Find the area of region between parabolas $y^2 = 4ax$ and $x^2 = 4ay$.Preview
  5. Q5Find the area of the region bounded by the curve $y^2=4x$, the X-axis and the lines $x=1$, $x=4$ for $y \ge 0$.Preview
  6. Q6Find the area of the region bounded by the curve $y=x^2$ and the line $y=4$.Preview
  7. Q7Find the area of the region bounded by the curve $y=x^2$, and the lines $x=1, x=2$ and $y=0$.Preview
  8. Q8The area bounded by the line $y=x$, X-axis and the lines $x=-1$ and $x=4$ is equal to ____. (in square units) (a) $\dfrac{2}{17}$ (b) 8 (c)…Preview
  9. Q9Find the area of the region bounded by the parabola $y^2=16x$ and its latus rectum.Preview
  10. Q10Find the area enclosed between the circle $x^2+y^2=1$ and the line $x+y=1$ lying in the first quadrant.Preview

More questions

52 Q
+Show 14 questions14 questions
  1. Q1Find the area of the region bounded by the following curves, X-axis and the given lines: $y = 2x,\ x = 0,\ x = 5$Free
  2. Q2Find the area of the region bounded by the following curves, X-axis and the given lines: $x = 2y,\ y = 0,\ y = 4$Free
  3. Q3Find the area of the region bounded by the following curves, X-axis and the given lines: $x = 0,\ x = 5,\ y = 0,\ y = 4$Free
  4. Q4Find the area of the region bounded by the following curves, X-axis and the given lines: $y = \sin x,\ x = 0,\ x = \dfrac{\pi}{2}$Preview
  5. Q5Find the area of the region bounded by the following curves, X-axis and the given lines: $xy = 2,\ x = 1,\ x = 4$Preview
  6. Q6Find the area of the region bounded by the following curves, X-axis and the given lines: $y^2 = x,\ x = 0,\ x = 4$Preview
  7. Q7Find the area of the region bounded by the following curves, X-axis and the given lines: $y^2 = 16x$ and $x = 0,\ x = 4$Preview
  8. Q8Find the area of the region bounded by the parabola: $y^2 = 16x$ and its latus rectum.Preview
  9. Q9Find the area of the region bounded by the parabola: $y = 4 - x^2$ and the X-axisPreview
  10. Q10Find the area of the region included between: $y^2 = 2x$, line $y = 2x$Preview
  11. Q11Find the area of the region included between: $y^2 = 4x$, line $y = x$Preview
  12. Q12Find the area of the region included between: $y = x^2$ and the line $y = 4x$Preview
  13. Q13Find the area of the region included between: $y^2 = 4ax$ and the line $y = x$Preview
  14. Q14Find the area of the region included between: $y = x^2 + 3$ and the line $y = x + 3$Preview
+Show 20 questions20 questions
  1. Q19The area bounded by the region $1 \le x \le 5$ and $2 \le y \le 5$ is given by (A) 12 sq. units (B) 8 sq. units (C) 25 sq. units (D) 32 sq.…Free
  2. Q20The area of the region enclosed by the curve $y = \dfrac{1}{x}$, and the lines $x = e,\ x = e^2$ is given by (A) 1 sq. unit (B) $\dfrac{1}{2…Free
  3. Q21The area bounded by the curve $y = x^3$, the X-axis and the lines $x = -2$ and $x = 1$ is (A) $-9$ sq. units (B) $-\dfrac{15}{4}$ sq. units…Free
  4. Q22The area enclosed between the parabola $y^2 = 4x$ and line $y = 2x$ is (A) $\dfrac{2}{3}$ sq. units (B) $\dfrac{1}{3}$ sq. units (C) $\dfrac…Preview
  5. Q23The area of the region bounded between the line $x = 4$ and the parabola $y^2 = 16x$ is (A) $\dfrac{128}{3}$ sq. units (B) $\dfrac{108}{3}$…Preview
  6. Q24The area of the region bounded by $y = \cos x$, Y-axis and the lines $x = 0,\ x = 2\pi$ is (A) 1 sq. unit (B) 2 sq. units (C) 3 sq. units (D…Preview
  7. Q25The area bounded by the parabola $y^2 = 8x$ the X-axis and the latus rectum is (A) $\dfrac{31}{3}$ sq. units (B) $\dfrac{32}{3}$ sq. units (…Preview
  8. Q26The area under the curve $y = 2\sqrt{x}$, enclosed between the lines $x = 0$ and $x = 1$ is (A) 4 sq. units (B) $\dfrac{3}{4}$ sq. units (C)…Preview
  9. Q27The area of the circle $x^2 + y^2 = 25$ in first quadrant is (A) $\dfrac{25\pi}{3}$ sq. units (B) $5\pi$ sq. units (C) 5 sq. units (D) 3 sq.…Preview
  10. Q28The area of the region bounded by the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ is (A) $ab$ sq. units (B) $\pi ab$ sq. units (C) $\d…Preview
  11. Q29The area bounded by the parabola $y^2 = x$ and the line $2y = x$ is (A) $\dfrac{4}{3}$ sq. units (B) 1 sq. units (C) $\dfrac{2}{3}$ sq. unit…Preview
  12. Q30The area enclosed between the curve $y = \cos 3x$, $0 \le x \le \dfrac{\pi}{6}$ and the X-axis is (A) $\dfrac{1}{2}$ sq. units (B) 1 sq. uni…Preview
  13. Q31The area bounded by $y = \sqrt{x}$ and line $x = 2y + 3$, X-axis in first quadrant is (A) $2\sqrt3$ sq. units (B) 9 sq. units (C) $\dfrac{34…Preview
  14. Q32The area bounded by the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ and the line $\dfrac{x}{a} + \dfrac{y}{b} = 1$ is (A) $\pi ab - 2a…Preview
  15. Q33The area bounded by the parabola $y = x^2$ and the line $y = x$ is (A) $\dfrac{1}{2}$ (B) $\dfrac{1}{3}$ (C) $\dfrac{1}{6}$ (D) $\dfrac{1}{1…Preview
  16. Q34The area enclosed between the two parabolas $y^2 = 4x$ and $y = x$ is (A) $\dfrac{8}{3}$ (B) $\dfrac{32}{3}$ (C) $\dfrac{16}{3}$ (D) $\dfrac…Preview
  17. Q35The area bounded by the curve $y = \tan x$, X-axis and the line $x = \dfrac{\pi}{4}$ is (A) $\dfrac{1}{3}\log 2$ (B) $\log 2$ (C) $2\log 2$…Preview
  18. Q36The area of the region bounded by $x^2 = 16y,\ y = 1,\ y = 4$ and $x = 0$ in the first quadrant, is (A) $\dfrac{7}{3}$ (B) $\dfrac{8}{3}$ (C…Preview
  19. Q37The area of the region included between the parabolas $y^2 = 4ax$ and $x^2 = 4ay$, $(a > 0)$ is given by (A) $\dfrac{16a^2}{3}$ (B) $\dfrac{…Preview
  20. Q38The area of the region included between the line $x + y = 1$ and the circle $x^2 + y^2 = 1$ is (A) $\dfrac{\pi}{2} - 1$ (B) $\pi - 2$ (C) $\…Preview
+Show 14 questions14 questions
  1. Q39Find the area of the region bounded by the following curve, the X-axis and the given lines: $0 \le x \le 5,\ 0 \le y \le 2$Free
  2. Q40Find the area of the region bounded by the following curve, the X-axis and the given lines: $y = \sin x,\ x = 0,\ x = \pi$Free
  3. Q41Find the area of the region bounded by the following curve, the X-axis and the given lines: $y = \sin x,\ x = 0,\ x = \dfrac{\pi}{3}$Free
  4. Q42Find the area of the circle $x^2 + y^2 = 9$, using integration.Preview
  5. Q43Find the area of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$ using integration.Preview
  6. Q44Find the area of the region lying between the parabolas: $y^2 = 4x$ and $x^2 = 4y$Preview
  7. Q45Find the area of the region lying between the parabolas: $4y^2 = 9x$ and $3x^2 = 16y$Preview
  8. Q46Find the area of the region lying between the parabolas: $y^2 = x$ and $x^2 = y$Preview
  9. Q47Find the area of the region in first quadrant bounded by the circle $x^2 + y^2 = 4$ and the x axis and the line $x = y\sqrt3$.Preview
  10. Q48Find the area of the region bounded by the parabola $y^2 = x$ and the line $y = x$ in the first quadrant.Preview
  11. Q49Find the area enclosed between the circle $x^2 + y^2 = 1$ and the line $x + y = 1$, lying in the first quadrant.Preview
  12. Q50Find the area of the region bounded by the curve $(y - 1)^2 = 4(x + 1)$ and the line $y = (x - 1)$.Preview
  13. Q51Find the area of the region bounded by the straight line $2y = 5x + 7$, X-axis and $x = 2,\ x = 5$.Preview
  14. Q52Find the area of the region bounded by the curve $y = 4x^2$, Y-axis and the lines $y = 1,\ y = 4$.Preview
+Show 4 questions4 questions
  1. Q15Find the area enclosed between $y = \sin x$ and X-axis between $0$ and $4\pi$.Free
  2. Q16Find the area enclosed between $y = \cos x$ and X-axis between the limits: $0 \le x \le \dfrac{\pi}{2}$Free
  3. Q17Find the area enclosed between $y = \cos x$ and X-axis between the limits: $\dfrac{\pi}{2} \le x \le \pi$Preview
  4. Q18Find the area enclosed between $y = \cos x$ and X-axis between the limits: $0 \le x \le \pi$Preview