Mathematics · Class 12 Science
Ch 2Application 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.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
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 .
Most relevant Q&A
- Find the area of the region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$.Free
- Find the area of the region bounded by the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$.Free
- Find 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
- The 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
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
Elementary geometry gives us formulas for the area of figures with straight edges or circular boundaries — triangles, rectangles, trapeziums, circles.
Area Under Simple Curves
6 QThe definite integral now finds a natural geometric use: computing the area of a region bounded by a curve and straight lines.
+−Worked Examplesi2 questions
- 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
- 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
- Q1Find the area of the region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$.Free
- Q2Find the area of the region bounded by the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$.Free
- 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
- 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
- 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
- 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
- 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
- Q2Sketch the graph of $y = |x+3|$ and evaluate $\int_{-6}^{0} |x+3| dx$.Free
- Q3Find the area bounded by the curve $y = \sin x$ between $x = 0$ and $x = 2\pi$.Preview
- 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
- 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 questionsHide questions19 questions
- Q1Find the area of the region bounded by the line $x = 2$ and the parabola $y^2 = 8x$.Free
- Q2Sketch the region $\{(x, 0) : y = \sqrt{4 - x^2}\}$ and x-axis. Find the area of the region using integration.Free
- Q3Calculate the area under the curve $y = 2\sqrt{x}$ included between the lines $x = 0$ and $x = 1$.Free
- 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
- 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
- Q6Determine the area under the curve $y = \sqrt{a^2 - x^2}$ included between the lines $x = 0$ and $x = a$.Preview
- Q7Find the area of the region bounded by the triangle whose vertices are $(-1, 1)$, $(0, 5)$ and $(3, 2)$, using integration.Preview
- Q8Compute the area bounded by the lines $x + 2y = 2$, $y - x = 1$ and $2x + y = 7$.Preview
- Q9Find the area bounded by the lines $y = 4x + 5$, $y = 5 - x$ and $4y = x + 5$.Preview
- Q10Find the area bounded by the curve $y = 2\cos x$ and the x-axis from $x = 0$ to $x = 2\pi$.Preview
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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.