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# RD Sharma solutions for Class 12 Maths chapter 21 - Areas of Bounded Regions [Latest edition]

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## Chapter 21: Areas of Bounded Regions

Ex. 21.1Ex. 21.2Ex. 21.3Ex. 21.4MCQ

#### RD Sharma solutions for Class 12 Maths Chapter 21 Areas of Bounded Regions Exercise 21.1 [Pages 14 - 16]

Ex. 21.1 | Q 1 | Page 14

Using integration, find the area of the region bounded between the line x = 2 and the parabola y2 = 8x.

Ex. 21.1 | Q 2 | Page 14

Using integration, find the area of the region bounded by the line y − 1 = x, the x − axis and the ordinates x= −2 and x = 3.

Ex. 21.1 | Q 3 | Page 15

Find the area of the region bounded by the parabola y2 = 4ax and the line x = a.

Ex. 21.1 | Q 4 | Page 15

Find the area lying above the x-axis and under the parabola y = 4x − x2.

Ex. 21.1 | Q 5 | Page 15

Draw a rough sketch to indicate the region bounded between the curve y2 = 4x and the line x = 3. Also, find the area of this region.

Ex. 21.1 | Q 6 | Page 15

Make a rough sketch of the graph of the function y = 4 − x2, 0 ≤ x ≤ 2 and determine the area enclosed by the curve, the x-axis and the lines x = 0 and x = 2.

Ex. 21.1 | Q 7 | Page 15

Sketch the graph of y = $\sqrt{x + 1}$  in [0, 4] and determine the area of the region enclosed by the curve, the x-axis and the lines x = 0, x = 4.

Ex. 21.1 | Q 8 | Page 15

Find the area under the curve y = $\sqrt{6x + 4}$ above x-axis from x = 0 to x = 2. Draw a sketch of curve also.

Ex. 21.1 | Q 9 | Page 15

Draw the rough sketch of y2 + 1 = x, x ≤ 2. Find the area enclosed by the curve and the line x = 2.

Ex. 21.1 | Q 10 | Page 15

Draw a rough sketch of the graph of the curve $\frac{x^2}{4} + \frac{y^2}{9} = 1$  and evaluate the area of the region under the curve and above the x-axis.

Ex. 21.1 | Q 11 | Page 15

Sketch the region {(x, y) : 9x2 + 4y2 = 36} and find the area of the region enclosed by it, using integration.

Ex. 21.1 | Q 12 | Page 15

Draw a rough sketch of the graph of the function y = 2 $\sqrt{1 - x^2}$ , x ∈ [0, 1] and evaluate the area enclosed between the curve and the x-axis.

Ex. 21.1 | Q 13 | Page 15

Determine the area under the curve y = $\sqrt{a^2 - x^2}$  included between the lines x = 0 and x = a.

Ex. 21.1 | Q 14 | Page 15

Using integration, find the area of the region bounded by the line 2y = 5x + 7, x-axis and the lines x = 2 and x = 8.

Ex. 21.1 | Q 15 | Page 15

Using definite integrals, find the area of the circle x2 + y2 = a2.

Ex. 21.1 | Q 16 | Page 15

Using integration, find the area of the region bounded by the following curves, after making a rough sketch: y = 1 + | x + 1 |, x = −2, x = 3, y = 0.

Ex. 21.1 | Q 17 | Page 15

Sketch the graph y = | x − 5 |. Evaluate $\int\limits_0^1 \left| x - 5 \right| dx$. What does this value of the integral represent on the graph.

Ex. 21.1 | Q 18 | Page 15

Sketch the graph y = | x + 3 |. Evaluate $\int\limits_{- 6}^0 \left| x + 3 \right| dx$. What does this integral represent on the graph?

Ex. 21.1 | Q 19 | Page 15

Sketch the graph y = | x + 1 |. Evaluate$\int\limits_{- 4}^2 \left| x + 1 \right| dx$. What does the value of this integral represent on the graph?

Ex. 21.1 | Q 20 | Page 15

Find the area of the region bounded by the curve xy − 3x − 2y − 10 = 0, x-axis and the lines x = 3, x = 4.

Ex. 21.1 | Q 21 | Page 15

Draw a rough sketch of the curve y = $\frac{\pi}{2} + 2 \sin^2 x$ and find the area between x-axis, the curve and the ordinates x = 0, x = π.

Ex. 21.1 | Q 22 | Page 15

Draw a rough sketch of the curve $y = \frac{x}{\pi} + 2 \sin^2 x$ and find the area between the x-axis, the curve and the ordinates x = 0 and x = π.

Ex. 21.1 | Q 23 | Page 16

Find the area bounded by the curve y = cos x, x-axis and the ordinates x = 0 and x = 2π.

Ex. 21.1 | Q 24 | Page 16

Show that the areas under the curves y = sin x and y = sin 2x between x = 0 and x =$\frac{\pi}{3}$  are in the ratio 2 : 3.

Ex. 21.1 | Q 25 | Page 16

Compare the areas under the curves y = cos2 x and y = sin2 x between x = 0 and x = π.

Ex. 21.1 | Q 26 | Page 16

Find the area bounded by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$  and the ordinates x = ae and x = 0, where b2 = a2 (1 − e2) and e < 1.

Ex. 21.1 | Q 27 | Page 16

Find the area of the minor segment of the circle $x^2 + y^2 = a^2$ cut off by the line $x = \frac{a}{2}$

Ex. 21.1 | Q 28 | Page 16

Find the area of the region bounded by the curve $x = a t^2 , y = 2\text{ at }$between the ordinates corresponding t = 1 and t = 2.

Ex. 21.1 | Q 29 | Page 16

Find the area enclosed by the curve x = 3cost, y = 2sin t.

#### RD Sharma solutions for Class 12 Maths Chapter 21 Areas of Bounded Regions Exercise 21.2 [Page 24]

Ex. 21.2 | Q 1 | Page 24

Find the area of the region in the first quadrant bounded by the parabola y = 4x2 and the lines x = 0, y = 1 and y = 4.

Ex. 21.2 | Q 2 | Page 24

Find the area of the region bounded by x2 = 16y, y = 1, y = 4 and the y-axis in the first quadrant.

Ex. 21.2 | Q 3 | Page 24

Find the area of the region bounded by x2 = 4ay and its latusrectum.

Ex. 21.2 | Q 4 | Page 24

Find the area of the region bounded by x2 + 16y = 0 and its latusrectum.

Ex. 21.2 | Q 5 | Page 24

Find the area of the region bounded by the curve $a y^2 = x^3$, the y-axis and the lines y = a and y = 2a.

#### RD Sharma solutions for Class 12 Maths Chapter 21 Areas of Bounded Regions Exercise 21.3 [Pages 51 - 53]

Ex. 21.3 | Q 1 | Page 51

Calculate the area of the region bounded by the parabolas y2 = x and x2 = y.

Ex. 21.3 | Q 2 | Page 51

Find the area of the region common to the parabolas 4y2 = 9x and 3x2 = 16y.

Ex. 21.3 | Q 3 | Page 51

Find the area of the region bounded by y =$\sqrt{x}$ and y = x.

Ex. 21.3 | Q 4 | Page 51

Find the area bounded by the curve y = 4 − x2 and the lines y = 0, y = 3.

Ex. 21.3 | Q 5 | Page 51

Find the area of the region $\left\{ \left( x, y \right): \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \leq \frac{x}{a} + \frac{y}{b} \right\}$

Ex. 21.3 | Q 6 | Page 51

Using integration, find the area of the region bounded by the triangle whose vertices are (2, 1), (3, 4) and (5, 2).

Ex. 21.3 | Q 7 | Page 51

Using integration, find the area of the region bounded by the triangle ABC whose vertices A, B, C are (−1, 1), (0, 5) and (3, 2) respectively.

Ex. 21.3 | Q 8 | Page 51

Using integration, find the area of the triangular region, the equations of whose sides are y = 2x + 1, y = 3x+ 1 and x = 4.

Ex. 21.3 | Q 9 | Page 51

Find the area of the region {(x, y) : y2 ≤ 8x, x2 + y2 ≤ 9}.

Ex. 21.3 | Q 10 | Page 51

Find the area of the region common to the circle x2 + y2 = 16 and the parabola y2 = 6x.

Ex. 21.3 | Q 11 | Page 51

Find the area of the region between the circles x2 + y2 = 4 and (x − 2)2 + y2 = 4.

Ex. 21.3 | Q 12 | Page 51

Find the area of the region included between the parabola y2 = x and the line x + y = 2.

Ex. 21.3 | Q 13 | Page 51

Draw a rough sketch of the region {(x, y) : y2 ≤ 3x, 3x2 + 3y2 ≤ 16} and find the area enclosed by the region using method of integration.

Ex. 21.3 | Q 14 | Page 51

Draw a rough sketch of the region {(x, y) : y2 ≤ 5x, 5x2 + 5y2 ≤ 36} and find the area enclosed by the region using method of integration.

Ex. 21.3 | Q 15 | Page 51

Draw a rough sketch and find the area of the region bounded by the two parabolas y2 = 4x and x2 = 4y by using methods of integration.

Ex. 21.3 | Q 16 | Page 51

Find the area included between the parabolas y2 = 4ax and x2 = 4by.

Ex. 21.3 | Q 17 | Page 51

Prove that the area in the first quadrant enclosed by the x-axis, the line x = $\sqrt{3}y$ and the circle x2 + y2 = 4 is π/3.

Ex. 21.3 | Q 18 | Page 51

Find the area of the region bounded by $y = \sqrt{x}, x = 2y + 3$  in the first quadrant and x-axis.

Ex. 21.3 | Q 19 | Page 51

Find the area common to the circle x2 + y2 = 16 a2 and the parabola y2 = 6 ax.
OR
Find the area of the region {(x, y) : y2 ≤ 6ax} and {(x, y) : x2 + y2 ≤ 16a2}.

Ex. 21.3 | Q 20 | Page 51

Find the area, lying above x-axis and included between the circle x2 + y2 = 8x and the parabola y2 = 4x.

Ex. 21.3 | Q 21 | Page 51

Find the area enclosed by the parabolas y = 5x2 and y = 2x2 + 9.

Ex. 21.3 | Q 22 | Page 52

Prove that the area common to the two parabolas y = 2x2 and y = x2 + 4 is $\frac{32}{3}$ sq. units.

Ex. 21.3 | Q 23 | Page 52

Using integration, find the area of the region bounded by the triangle whose vertices are (−1, 2), (1, 5) and (3, 4).

Ex. 21.3 | Q 24 | Page 52

Find the area of the region bounded by $y = \sqrt{x}$ and y = x.

Ex. 21.3 | Q 25 | Page 52

Find the area of the region in the first quadrant enclosed by x-axis, the line y = $\sqrt{3}x$ and the circle x2 + y2 = 16.

Ex. 21.3 | Q 26 | Page 52

Find the area of the region bounded by the parabola y2 = 2x + 1 and the line x − y − 1 = 0.

Ex. 21.3 | Q 27 | Page 52

Find the area of the region bounded by the curves y = x − 1 and (y − 1)2 = 4 (x + 1).

Ex. 21.3 | Q 28 | Page 52

Find the area enclosed by the curve $y = - x^2$ and the straight line x + y + 2 = 0.

Ex. 21.3 | Q 29 | Page 52

Find the area bounded by the parabola y = 2 − x2 and the straight line y + x = 0.

Ex. 21.3 | Q 30 | Page 52

Using the method of integration, find the area of the region bounded by the following lines:
3x − y − 3 = 0, 2x + y − 12 = 0, x − 2y − 1 = 0.

Ex. 21.3 | Q 31 | Page 52

Sketch the region bounded by the curves y = x2 + 2, y = x, x = 0 and x = 1. Also, find the area of this region.

Ex. 21.3 | Q 32 | Page 52

Find the area bounded by the curves x = y2 and x = 3 − 2y2.

Ex. 21.3 | Q 33 | Page 52

Using integration, find the area of the triangle ABC coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4).

Ex. 21.3 | Q 34 | Page 52

Using integration find the area of the region:
$\left\{ \left( x, y \right) : \left| x - 1 \right| \leq y \leq \sqrt{5 - x^2} \right\}$

Ex. 21.3 | Q 35 | Page 52

Find the area of the region bounded by y = | x − 1 | and y = 1.

Ex. 21.3 | Q 36 | Page 52

Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2= 32.

Ex. 21.3 | Q 37 | Page 52

Find the area of the circle x2 + y2 = 16 which is exterior to the parabola y2 = 6x.

Ex. 21.3 | Q 38 | Page 52

Find the area of the region enclosed by the parabola x2 = y and the line y = x + 2.

Ex. 21.3 | Q 39 | Page 52

Make a sketch of the region {(x, y) : 0 ≤ y ≤ x2 + 3; 0 ≤ y ≤ 2x + 3; 0 ≤ x ≤ 3} and find its area using integration.

Ex. 21.3 | Q 40 | Page 52

Find the area of the region bounded by the curve y = $\sqrt{1 - x^2}$, line y = x and the positive x-axis.

Ex. 21.3 | Q 41 | Page 52

Find the area bounded by the lines y = 4x + 5, y = 5 − x and 4y = x + 5.

Ex. 21.3 | Q 42 | Page 52

Find the area of the region enclosed between the two curves x2 + y2 = 9 and (x − 3)2 + y2 = 9.

Ex. 21.3 | Q 43 | Page 52

Find the area of the region {(x, y): x2 + y2 ≤ 4, x + y ≥ 2}.

Ex. 21.3 | Q 44 | Page 52

Using integration, find the area of the following region: $\left\{ \left( x, y \right) : \frac{x^2}{9} + \frac{y^2}{4} \leq 1 \leq \frac{x}{3} + \frac{y}{2} \right\}$

Ex. 21.3 | Q 45 | Page 53

Using integration find the area of the region bounded by the curves $y = \sqrt{4 - x^2}, x^2 + y^2 - 4x = 0$ and the x-axis.

Ex. 21.3 | Q 46 | Page 53

Find the area enclosed by the curves y = | x − 1 | and y = −| x − 1 | + 1.

Ex. 21.3 | Q 47 | Page 53

Find the area enclosed by the curves 3x2 + 5y = 32 and y = | x − 2 |.

Ex. 21.3 | Q 48 | Page 53

Find the area enclosed by the parabolas y = 4x − x2 and y = x2 − x.

Ex. 21.3 | Q 49 | Page 53

In what ratio does the x-axis divide the area of the region bounded by the parabolas y = 4x − x2 and y = x2− x?

Ex. 21.3 | Q 50 | Page 53

Find the area of the figure bounded by the curves y = | x − 1 | and y = 3 −| x |.

Ex. 21.3 | Q 51 | Page 53

If the area bounded by the parabola $y^2 = 4ax$ and the line y = mx is $\frac{a^2}{12}$ sq. units, then using integration, find the value of m.

Ex. 21.3 | Q 52 | Page 53

If the area enclosed by the parabolas y2 = 16ax and x2 = 16ay, a > 0 is $\frac{1024}{3}$ square units, find the value of a.

#### RD Sharma solutions for Class 12 Maths Chapter 21 Areas of Bounded Regions Exercise 21.4 [Page 61]

Ex. 21.4 | Q 1 | Page 61

Find the area of the region between the parabola x = 4y − y2 and the line x = 2y − 3.

Ex. 21.4 | Q 2 | Page 61

Find the area bounded by the parabola x = 8 + 2y − y2; the y-axis and the lines y = −1 and y = 3.

Ex. 21.4 | Q 3.1 | Page 61

Find the area bounded by the parabola y2 = 4x and the line y = 2x − 4 By using horizontal strips.

Ex. 21.4 | Q 3.2 | Page 61

Find the area bounded by the parabola y2 = 4x and the line y = 2x − 4 By using vertical strips.

Ex. 21.4 | Q 4 | Page 61

Find the area of the region bounded by the parabola y2 = 2x and the straight line x − y = 4.

#### RD Sharma solutions for Class 12 Maths Chapter 21 Areas of Bounded Regions Exercise MCQ [Pages 62 - 64]

MCQ | Q 1 | Page 62

If the area above the x-axis, bounded by the curves y = 2kx and x = 0, and x = 2 is $\frac{3}{\log_e 2}$, then the value of k is __________ .

• 1/2

• 1

• -1

• 2

MCQ | Q 2 | Page 62

The area included between the parabolas y2 = 4x and x2 = 4y is (in square units)

• 4/3

• 1/3

• 16/3

• 8/3

MCQ | Q 3 | Page 62

The area bounded by the curve y = loge x and x-axis and the straight line x = e is ___________ .

• e sq. units

• 1 sq. units

• 1−$\frac{1}{e}$ sq. units

• 1+$\frac{1}{e}$ sq. units

MCQ | Q 4 | Page 62

The area bounded by y = 2 − x2 and x + y = 0 is _________ .

• $\frac{7}{2}$ sq. units

• $\frac{9}{2}$ sq. units

• 9 sq. units

• none of these

MCQ | Q 5 | Page 62

The area bounded by the parabola x = 4 − y2 and y-axis, in square units, is ____________ .

• $\frac{3}{32}$

• $\frac{32}{3}$

• $\frac{33}{2}$

• $\frac{16}{3}$

MCQ | Q 6 | Page 62

If An be the area bounded by the curve y = (tan x)n and the lines x = 0, y = 0 and x = π/4, then for x > 2

• An + An −2 = $\frac{1}{n - 1}$

• An + An − 2 < $\frac{1}{n - 1}$

• An − An − 2 = $\frac{1}{n - 1}$

• none of these

MCQ | Q 7 | Page 62

The area of the region formed by x2 + y2 − 6x − 4y + 12 ≤ 0, y ≤ x and x ≤ 5/2 is ______ .

• $\frac{\pi}{6} - \frac{\sqrt{3} + 1}{8}$

• $\frac{\pi}{6} + \frac{\sqrt{3} + 1}{8}$

• $\frac{\pi}{6} - \frac{\sqrt{3} - 1}{8}$

• none of these

MCQ | Q 8 | Page 62

The area enclosed between the curves y = loge (x + e), x = log$\left( \frac{1}{y} \right)$ and the x-axis is _______ .

• 2

• 1

• 4

• none of these

MCQ | Q 9 | Page 62

The area of the region bounded by the parabola (y − 2)2 = x − 1, the tangent to it at the point with the ordinate 3 and the x-axis is _________ .

• 3

• 6

• 7

• none of these

MCQ | Q 10 | Page 62

The area bounded by the curves y = sin x between the ordinates x = 0, x = π and the x-axis is _____________ .

• 2 sq. units

• 4 sq. units

• 3 sq. units

• 1 sq. unit

MCQ | Q 11 | Page 62

The area bounded by the parabola y2 = 4ax and x2 = 4ay is ___________ .

• $\frac{8 a^3}{3}$

• $\frac{16 a^2}{3}$

• $\frac{32 a^2}{3}$

• $\frac{64 a^2}{3}$

MCQ | Q 12 | Page 62

The area bounded by the curve y = x4 − 2x3 + x2 + 3 with x-axis and ordinates corresponding to the minima of y is _________ .

• 1

• $\frac{91}{30}$

• $\frac{30}{9}$

• 4

MCQ | Q 13 | Page 63

The area bounded by the parabola y2 = 4ax, latusrectum and x-axis is ___________ .

• 0

• $\frac{4}{3} a^2$

• $\frac{2}{3} a^2$

• $\frac{a^2}{3}$

MCQ | Q 14 | Page 63

The area of the region $\left\{ \left( x, y \right) : x^2 + y^2 \leq 1 \leq x + y \right\}$ is __________ .

• $\frac{\pi}{5}$

• $\frac{\pi}{4}$

• $\frac{\pi}{2} - \frac{1}{2}$

• $\frac{\pi^2}{2}$

• None of these

MCQ | Q 15 | Page 63

The closed area made by the parabola y = 2x2 and y = x2 + 4 is __________ .

• $\frac{2}{3}$sq. units

• $\frac{3}{2}$sq. units

• $\frac{32}{3}$sq. units

• $\frac{3}{32}$sq. units

MCQ | Q 16 | Page 63

The area of the region bounded by the parabola y = x2 + 1 and the straight line x + y = 3 is given by

• $\frac{45}{7}$

• $\frac{25}{4}$

• $\frac{\pi}{18}$

• $\frac{9}{2}$

MCQ | Q 17 | Page 63

The ratio of the areas between the curves y = cos x and y = cos 2x and x-axis from x = 0 to x = π/3 is ________ .

• 1 : 2

• 2 : 1

• $\sqrt{3}$

• none of these

MCQ | Q 18 | Page 63

The area between x-axis and curve y = cos x when 0 ≤ x ≤ 2 π is ___________ .

• 0

• 2

• 3

• 4

MCQ | Q 19 | Page 63

Area bounded by parabola y2 = x and straight line 2y = x is _________ .

• 4/3

• 1

• 2/3

• 1/3

MCQ | Q 20 | Page 63

The area bounded by the curve y = 4x − x2 and the x-axis is __________ .

• $\frac{30}{7}$sq. units

• $\frac{31}{7}$sq. units

• $\frac{32}{3}$sq. units

• $\frac{34}{3}$sq. units

MCQ | Q 21 | Page 63

Area enclosed between the curve y2 (2a − x) = x3 and the line x = 2a above x-axis is ___________ .

• πa2

• $\frac{3}{2}\pi a^2$

• 2πa2

• 3πa2

MCQ | Q 22 | Page 63

The area of the region (in square units) bounded by the curve x2 = 4y, line x = 2 and x-axis is

• 1

• 2/3

• 4/3

• 8/3

MCQ | Q 23 | Page 63

The area bounded by the curve y = f (x), x-axis, and the ordinates x = 1 and x = b is (b −1) sin (3b + 4). Then, f (x) is __________ .

• (x − 1) cos (3x + 4)

• sin (3x + 4)

• sin (3x + 4) + 3 (x − 1) cos (3x +4)

• none of these

MCQ | Q 24 | Page 63

The area bounded by the curve y2 = 8x and x2 = 8y is ___________ .

• $\frac{16}{3}$sq. units

• $\frac{3}{16}$sq. units

• $\frac{14}{3}$sq. units

• $\frac{3}{14}$sq. units

• None of these

MCQ | Q 25 | Page 63

The area bounded by the parabola y2 = 8x, the x-axis and the latusrectum is ___________ .

• $\frac{16}{3}$
• $\frac{23}{3}$
• $\frac{32}{3}$
• $\frac{16\sqrt{2}}{3}$
MCQ | Q 26 | Page 63

Area bounded by the curve y = x3, the x-axis and the ordinates x = −2 and x = 1 is __________ .

• −9

• $\frac{- 15}{4}$
• $\frac{15}{4}$
• $\frac{17}{4}$
MCQ | Q 27 | Page 64

The area bounded by the curve y = x |x| and the ordinates x = −1 and x = 1 is given by

• 0

• $\frac{1}{3}$
• $\frac{2}{3}$
• $\frac{4}{3}$
MCQ | Q 28 | Page 64

The area bounded by the y-axis, y = cos x and y = sin x when 0 ≤ x ≤ $\frac{\pi}{2}$ is _________ .

• 2$\left( \sqrt{2} - 1 \right)$

• $\sqrt{2} - 1$
• $\sqrt{2} + 1$
• $\sqrt{2}$
MCQ | Q 29 | Page 64

The area of the circle x2 + y2 = 16 enterior to the parabola y2 = 6x is

• $\frac{4}{3}\left( 4\pi - \sqrt{3} \right)$
• $\frac{4}{3}\left( 4\pi + \sqrt{3} \right)$
• $\frac{4}{3}\left( 8\pi - \sqrt{3} \right)$
• $\frac{4}{3}\left( 8\pi + \sqrt{3} \right)$
MCQ | Q 30 | Page 64

Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is

• 2 (π − 2)

• π − 2

• 2π − 1

• 2 (π + 2)

MCQ | Q 31 | Page 64

Area lying between the curves y2 = 4x and y = 2x is

• $\frac{2}{3}$
• $\frac{1}{3}$
• $\frac{1}{4}$
• $\frac{3}{4}$
MCQ | Q 32 | Page 64

Area lying in first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2, is

• π

• $\frac{\pi}{2}$
• $\frac{\pi}{3}$
• $\frac{\pi}{4}$
MCQ | Q 33 | Page 64

Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3, is

• 2

• $\frac{9}{4}$
• $\frac{9}{3}$
• $\frac{9}{2}$

## Chapter 21: Areas of Bounded Regions

Ex. 21.1Ex. 21.2Ex. 21.3Ex. 21.4MCQ

## RD Sharma solutions for Class 12 Maths chapter 21 - Areas of Bounded Regions

RD Sharma solutions for Class 12 Maths chapter 21 (Areas of Bounded Regions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Maths chapter 21 Areas of Bounded Regions are Area of the Region Bounded by a Curve and a Line, Area Between Two Curves, Area Under Simple Curves.

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