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Mathematics · Class 12 Science

Ch 8Application of Integrals — Class 12 Mathematics, concept-first.

Elementary geometry gives us formulas for the area of figures with straight edges or circular boundaries — triangles, rectangles, trapeziums, circles. These formulas are essential to countless real-life applications of mathematics, but they only go so far: they cannot handle a region whose boundary is a general curve.

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Key concepts

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Area of Ellipse

A circle of radius has area . An ellipse is a circle that has been stretched by different amounts along two perpendicular directions, so it is natural to expect its area to be a stretched version of .

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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.

8.1

Introduction

Elementary geometry gives us formulas for the area of figures with straight edges or circular boundaries — triangles, rectangles, trapeziums, circles.

8.2

Area Under Simple Curves

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The definite integral now finds a natural geometric use: computing the area of a region bounded by a curve and straight lines.

+Worked Examplesi2 questions
  1. Example 1Find the area enclosed by the circle $x^2 + y^2 = a^2$ ![Quarter of the circle x² + y² = a², shaded to show the area bounded by the circle a…Free
  2. Example 2Find the area enclosed by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ![Quarter of the ellipse x²/a² + y²/b² = 1, shaded to show the…Preview
+Exercise 8.1i4 questions
  1. Q1Find the area of the region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$.Free
  2. Q2Find the area of the region bounded by the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$.Free
  3. Q3Find the value of the following: Area lying in the first quadrant and bounded by the circle $x^2 + y^2 = 4$ and the lines $x = 0$ and $x = 2…Preview
  4. Q4Find the value of the following: Area of the region bounded by the curve $y^2 = 4x$, y-axis and the line $y = 3$ is (A) 2 (B) $\frac{9}{4}$…Preview

Miscellaneous Examples

+Miscellaneous Examplesi2 questions
  1. Example 3Find the area of the region bounded by the line $y = 3x + 2$, the x-axis and the ordinates $x = -1$ and $x = 1$ ![Graph of the line y = 3x +…Free
  2. Example 4Find the area bounded by the curve $y = \cos x$ between $x = 0$ and $x = 2\pi$ ![Graph of y = cos x from 0 to 2π, with the regions above and…Preview

Miscellaneous Exercise on Chapter 8

+Miscellaneous Exercisei5 questions
  1. Q1Find the area under the given curves and given lines: (i) $y = x^2, x = 1, x = 2$ and $x$-axis (ii) $y = x^4, x = 1, x = 5$ and $x$-axisFree
  2. Q2Sketch the graph of $y = |x+3|$ and evaluate $\int_{-6}^{0} |x+3| dx$.Free
  3. Q3Find the area bounded by the curve $y = \sin x$ between $x = 0$ and $x = 2\pi$.Preview
  4. Q4Find the value of the following: Area bounded by the curve $y = x^3$, the $x$-axis and the ordinates $x = -2$ and $x = 1$ is (A) $-9$ (B) $-…Preview
  5. Q5The area bounded by the curve $y = x|x|$, $x$-axis and the ordinates $x = -1$ and $x = 1$ is given by (A) $0$ (B) $\frac{1}{3}$ (C) $\frac{2…Preview

Summary

- Area under a curve: The area bounded by , the -axis, and vertical lines , is (if ). For , area is .

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 19 questions19 questions
  1. Q1Find the area of the region bounded by the line $x = 2$ and the parabola $y^2 = 8x$.Free
  2. Q2Sketch the region $\{(x, 0) : y = \sqrt{4 - x^2}\}$ and x-axis. Find the area of the region using integration.Free
  3. Q3Calculate the area under the curve $y = 2\sqrt{x}$ included between the lines $x = 0$ and $x = 1$.Free
  4. Q4Using integration, find the area of the region bounded by the line $2y = 5x + 7$, x-axis and the lines $x = 2$ and $x = 8$.Preview
  5. Q5Draw a rough sketch of the curve $y = \sqrt{x - 1}$ in the interval $[1, 5]$. Find the area under the curve and between the lines $x = 1$ an…Preview
  6. Q6Determine the area under the curve $y = \sqrt{a^2 - x^2}$ included between the lines $x = 0$ and $x = a$.Preview
  7. Q7Find the area of the region bounded by the triangle whose vertices are $(-1, 1)$, $(0, 5)$ and $(3, 2)$, using integration.Preview
  8. Q8Compute the area bounded by the lines $x + 2y = 2$, $y - x = 1$ and $2x + y = 7$.Preview
  9. Q9Find the area bounded by the lines $y = 4x + 5$, $y = 5 - x$ and $4y = x + 5$.Preview
  10. Q10Find the area bounded by the curve $y = 2\cos x$ and the x-axis from $x = 0$ to $x = 2\pi$.Preview
  11. Q11Draw a rough sketch of the given curve $y = 1 + |x + 1|$, $x = -3$, $x = 3$, $y = 0$ and find the area of the region bounded by them, using…Preview
  12. Q12The area of the region bounded by the curve $y = \sqrt{16 - x^2}$ and x-axis is (A) $8$ sq units (B) $20\pi$ sq units (C) $16\pi$ sq units (…Preview
  13. Q13Area of the region in the first quadrant enclosed by the x-axis, the line $y = x$ and the circle $x^2 + y^2 = 32$ is (A) $16\pi$ sq units (B…Preview
  14. Q14Area of the region bounded by the curve $y = \cos x$ between $x = 0$ and $x = \pi$ is (A) $2$ sq units (B) $4$ sq units (C) $3$ sq units (D)…Preview
  15. Q15The area of the region bounded by the curve $y = \sin x$ between the ordinates $x = 0$, $x = \frac{\pi}{2}$ and the x-axis is (A) $2$ sq uni…Preview
  16. Q16The area of the region bounded by the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ is (A) $20\pi$ sq units (B) $20\pi^2$ sq units (C) $16\p…Preview
  17. Q17The area of the region bounded by the circle $x^2 + y^2 = 1$ is (A) $2\pi$ sq units (B) $\pi$ sq units (C) $3\pi$ sq units (D) $4\pi$ sq uni…Preview
  18. Q18The area of the region bounded by the curve $y = x + 1$ and the lines $x = 2$ and $x = 3$ is (A) $\frac{7}{2}$ sq units (B) $\frac{9}{2}$ sq…Preview
  19. Q19The area of the region bounded by the curve $x = 2y + 3$ and the lines $y = 1$ and $y = -1$ is (A) $4$ sq units (B) $\frac{3}{2}$ sq units (…Preview

Sample & Board Papers

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

+Show 2 questions2 questions
  1. Q1To reduce traffic and avoid red lights, roundabouts are often built on busy roads. Such a roundabout is built such that its boundary $C_1 :…Preview
  2. Q2Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line $y = x$ and the circle $x^2 + y^2 = 32…Preview