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Mathematics · Class 12 Science

Ch 11Applications of the Integrals — Class 12 Mathematics, concept-first.

Until now, the definite integral has mostly been a number you compute by finding an antiderivative and applying the Fundamental Theorem of Calculus. This chapter gives that number a geometric meaning: when throughout , the definite integral equals the exact area of the region bounded above by the curve , below by the -…

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Concepts

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Unit weightage

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Key concepts

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Area between Two Curves

When two curves and cross each other, they enclose a bounded region between them. To find its area, the first step is always to locate the points where the two curves meet, by solving their equations simultaneously -- th…

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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.

1

The Area-Under-a-Curve Idea

Until now, the definite integral has mostly been a number you compute by finding an antiderivative and applying the Fundamental Theorem of Calculus.

2

Area Under a Straight Line

The simplest curve to which the area idea of Section 1 applies is a straight line . Suppose the line stays at or above the -axis for every in (with ).

3

Area of a Circle (Standard Form)

The standard equation of a circle of radius centred at the origin is . Solving for , the upper half of the circle is the curve and the full circle's area is derived by integrating this expression and…

4

Area Under a Parabola (Standard Form)

A parabola in standard form, such as (opening to the right, vertex at the origin, with ), is not a function of by itself -- solving for gives two branches, , symmetric about the -axis.

5

Area of an Ellipse (Standard Form)

The standard equation of an ellipse centred at the origin, with semi-axes (along ) and (along ), is . Solving for the upper half, Notice that this is exactly the circle's semicircle expression (a circ…

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Area Between Two Curves

Sections 2-5 each found the area under a single curve. Many exam questions instead ask for the area of a region enclosed between two curves -- most commonly a line and a parabola, or a line and an ell…

Summary

The core idea. For on , the definite integral equals the area bounded by , the -axis, and the ordinates -- justified by the Riemann-sum limit of thin rectangular strips.

Sample & Board Papers

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

+Show 8 questions8 questions
  1. Q1Using integral calculus, find the area of x^2/2 + y^2/1 = 1.Preview
  2. Q2Find the area of the circle x² + y² = 16 using integral calculus.Preview
  3. Q3Find the area bounded by y = x, x = 2 and x-axis. (using calculus)Preview
  4. Q4Find the area of circle x² + y² = 8x using calculus.Preview
  5. Q5Find the area of the triangle formed by y = 4x, x-axis and x = 4. (using Calculus)Preview
  6. Q6Find the area bounded by the curve y=sinx, x=0, x=π and the x-axis.Preview
  7. Q7Find the area of the region bounded by the curves $y = \sqrt{x}$, $2y - x + 3 = 0$, $x$-axis and lying in the first quadrant.Preview
  8. Q8Find the area in the first quadrant enclosed by the $x$-axis, the line $x = \sqrt{3}\,y$ and the circle $x^2 + y^2 = 4$, using definite inte…Preview

More questions

25 Q
+Show 3 questions3 questions
  1. Q23Find the area of the region bounded by the circle $x^2 + y^2 = 16$ and the line $x = 2$, lying to the right of the line $x = 2$.Free
  2. Q24Find the area of the region bounded by the line $y = 3x$, the $y$-axis, and the line $y = 6$.Preview
  3. Q25Find the whole area of the ellipse $9x^2 + 25y^2 = 225$, using integration.Preview
+Show 7 questions7 questions
  1. Example 1Find the area of the region bounded by the line $y = 3x + 1$, the $x$-axis, and the ordinates $x = 1$ and $x = 3$.Free
  2. Example 2Find the area of the region bounded by the circle $x^2 + y^2 = 25$ and the $x$-axis, lying above the $x$-axis.Free
  3. Example 3Find the area enclosed by the circle $x^2 + y^2 = 4$, using integration.Free
  4. Example 4Find the area of the region bounded by the parabola $y^2 = 8x$ and its latus rectum (the line $x = 2$).Preview
  5. Example 5Find the area of the region in the first quadrant bounded by the parabola $y^2 = 4x$, the $y$-axis, and the line $y = 4$.Preview
  6. Example 6Find the area of the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$, using integration.Preview
  7. Example 7Find the area of the region bounded by the curve $y = x^2$ and the line $y = x$.Preview
+Show 3 questions3 questions
  1. Q8Find the area of the region bounded by the line $y = 2x + 3$, the $x$-axis, and the ordinates $x = 0$ and $x = 2$.Free
  2. Q9Find the area of the triangle bounded by the line $y = x$, the $x$-axis, and the line $x = 4$, using integration.Preview
  3. Q10Find the area of the region bounded by the line $3x + 2y = 12$ and the coordinate axes, in the first quadrant.Preview
+Show 3 questions3 questions
  1. Q11Find the area enclosed by the circle $x^2 + y^2 = 36$, using integration.Free
  2. Q12Find the area of the region bounded by the circle $x^2 + y^2 = 16$ and the $x$-axis, lying above the $x$-axis.Preview
  3. Q13Find the area of the part of the circle $x^2 + y^2 = 9$ that lies in the first quadrant.Preview
+Show 3 questions3 questions
  1. Q14Find the area of the region bounded by the parabola $y^2 = 4x$ and the line $x = 4$.Free
  2. Q15Find the area of the region in the first quadrant bounded by the parabola $y^2 = 9x$, the $y$-axis, and the line $y = 6$.Preview
  3. Q16Find the area of the region bounded by the parabola $y^2 = 16x$ and its latus rectum.Preview
+Show 3 questions3 questions
  1. Q17Find the area of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$, using integration.Free
  2. Q18Find the area of the ellipse $4x^2 + 9y^2 = 36$, using integration.Preview
  3. Q19Find the area of the region in the first quadrant bounded by the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$, using integration.Preview
+Show 3 questions3 questions
  1. Q20Find the area of the region bounded by the parabola $y = x^2$ and the line $y = 4$.Free
  2. Q21Find the area of the region bounded by the parabola $x^2 = y$ and the line $y = x + 2$.Preview
  3. Q22Find the area of the smaller region bounded by the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ and the line $\dfrac{x}{3} + \dfrac{y}{2} =…Preview