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

## Chapter 21 : Areas of Bounded Regions

#### Pages 14 - 16

Q 1 | Page 14

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

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.

Q 3 | Page 15

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

Q 4 | Page 15

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

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.

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.

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.

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.

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.

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.

Q 11 | Page 15

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

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.

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.

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.

Q 15 | Page 15

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

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.

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.

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?

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?

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.

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.

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.

Q 23 | Page 16

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

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.

Q 25 | Page 16

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

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.

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 $x = \frac{a}{2}$

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.

Q 29 | Page 16

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

#### Page 24

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.

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.

Q 3 | Page 24

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

Q 4 | Page 24

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

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.

#### Pages 51 - 53

Q 1 | Page 51

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

Q 2 | Page 51

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

Q 3 | Page 51

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

Q 4 | Page 51

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

Q 5 | Page 51

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

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).

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.

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.

Q 9 | Page 51

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

Q 10 | Page 51

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

Q 11 | Page 51

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

Q 12 | Page 51

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

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.

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.

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.

Q 16 | Page 51

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

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.

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.

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}.

Q 20 | Page 51

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

Q 21 | Page 51

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

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.

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).

Q 24 | Page 52

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

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.

Q 26 | Page 52

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

Q 27 | Page 52

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

Q 28 | Page 52

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

Q 29 | Page 52

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

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.

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.

Q 32 | Page 52

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

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).

Q 34 | Page 52

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

Q 35 | Page 52

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

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.

Q 37 | Page 52

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

Q 38 | Page 52

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

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.

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.

Q 41 | Page 52

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

Q 42 | Page 52

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

Q 43 | Page 52

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

Q 44 | Page 52

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

Q 45 | Page 53

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

Q 46 | Page 53

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

Q 47 | Page 53

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

Q 48 | Page 53

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

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?

Q 50 | Page 53

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

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.

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.

#### Page 61

Q 1 | Page 61

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

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.

Q 3 | Page 61

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

Q 4 | Page 61

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

#### Pages 62 - 64

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
(a) 1/2
(b) 1
(c) −1
(d) 2

Q 2 | Page 62

The area included between the parabolas y2 = 4x and x2 = 4y is (in square units)
(a) 4/3
(b) 1/3
(c) 16/3
(d) 8/3

Q 3 | Page 62

The area bounded by the curve y = loge x and x-axis and the straight line x = e is
(a) e sq. units
(b) 1 sq. units
(c) 1−$\frac{1}{e}$ sq. units
(d) 1+$\frac{1}{e}$ sq. units

Q 4 | Page 62

The area bounded by y = 2 − x2 and x + y = 0 is
(a) $\frac{7}{2}$ sq. units
(b)$\frac{9}{2}$ sq. units
(c) 9 sq. units
(d) none of these

Q 5 | Page 62

The area bounded by the parabola x = 4 − y2 and y-axis, in square units, is
(a) $\frac{3}{32}$
(b) $\frac{32}{3}$
(c) $\frac{33}{2}$
(d) $\frac{16}{3}$

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
(a) An + An −2 = $\frac{1}{n - 1}$
(b) An + An − 2 < $\frac{1}{n - 1}$
(c) An − An − 2 = $\frac{1}{n - 1}$
(d) none of these

Q 7 | Page 62

The area of the region formed by x2 + y2 − 6x − 4y + 12 ≤ 0, y ≤ x and x ≤ 5/2 is
(a) $\frac{\pi}{6} - \frac{\sqrt{3} + 1}{8}$
(b) $\frac{\pi}{6} + \frac{\sqrt{3} + 1}{8}$
(c) $\frac{\pi}{6} - \frac{\sqrt{3} - 1}{8}$
(d) none of these

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
(a) 2
(b) 1
(c) 4
(d) none of these

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
(a) 3
(b) 6
(c) 7
(d) none of these

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
(a) 2 sq. units
(b) 4 sq. units
(c) 3 sq. units
(d) 1 sq. unit

Q 11 | Page 62

The area bounded by the parabola y2 = 4ax and x2 = 4ay is
(a) $\frac{8 a^3}{3}$
(b) $\frac{16 a^2}{3}$
(c) $\frac{32 a^2}{3}$
(d) $\frac{64 a^2}{3}$

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
(a) 1
(b) $\frac{91}{30}$
(c) $\frac{30}{9}$
(d) 4

Q 13 | Page 63

The area bounded by the parabola y2 = 4ax, latusrectum and x-axis is
(a) 0
(b) $\frac{4}{3} a^2$
(c) $\frac{2}{3} a^2$
(d) $\frac{a^2}{3}$

Q 14 | Page 63

The area of the region $\left\{ \left( x, y \right) : x^2 + y^2 \leq 1 \leq x + y \right\}$
(a) $\frac{\pi}{5}$
(b) $\frac{\pi}{4}$
(c) $\frac{\pi}{2} - \frac{1}{2}$
(d) $\frac{\pi^2}{2}$

Q 15 | Page 63

The area common to the parabola y = 2x2 and y = x2 + 4 is
(a) $\frac{2}{3}$sq. units
(b) $\frac{3}{2}$sq. units
(c) $\frac{32}{3}$sq. units
(d) $\frac{3}{32}$sq. units

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
(a) $\frac{45}{7}$
(b) $\frac{25}{4}$
(c) $\frac{\pi}{18}$
(d) $\frac{9}{2}$

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
(a) 1 : 2
(b) 2 : 1
(c) $\sqrt{3}$
(d) none of these

Q 18 | Page 63

The area between x-axis and curve y = cos x when 0 ≤ x ≤ 2 π is
(a) 0
(b) 2
(c) 3
(d) 4

Q 19 | Page 63

Area bounded by parabola y2 = x and straight line 2y = x is
(a) 43
(b) 1
(c) 23
(d) 13

Q 20 | Page 63

The area bounded by the curve y = 4x − x2 and the x-axis is
(a) $\frac{30}{7}$sq. units
(b) $\frac{31}{7}$sq. units
(c) $\frac{32}{3}$sq. units
(d) $\frac{34}{3}$sq. units

Q 21 | Page 63

Area enclosed between the curve y2 (2a − x) = x3 and the line x = 2a above x-axis is
(a) πa2
(b) $\frac{3}{2}\pi a^2$
(c) 2πa2
(d) 3πa2

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

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

Q 24 | Page 63

The area bounded by the curve y2 = 8x and x2 = 8y is
(a) $\frac{16}{3}$sq. units
(b) $\frac{3}{16}$sq. units
(c) $\frac{14}{3}$sq. units
(d) $\frac{3}{14}$sq. units

Q 25 | Page 63

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

$\frac{16}{3}$

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

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

−9

$\frac{- 15}{4}$

$\frac{15}{4}$
$\frac{17}{4}$
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

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{4}{3}$
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$\left( \sqrt{2} - 1 \right)$

$\sqrt{2} - 1$

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

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

$\frac{4}{3}\left( 4\pi - \sqrt{3} \right)$
$\frac{4}{3}\left( 4\pi + \sqrt{3} \right)$
$\frac{4}{3}\left( 8\pi - \sqrt{3} \right)$
$\frac{4}{3}\left( 8\pi + \sqrt{3} \right)$

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)

Q 31 | Page 64

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

$\frac{2}{3}$
$\frac{1}{3}$
$\frac{1}{4}$
$\frac{3}{4}$
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

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

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

2

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

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

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