# Solution of Chapter 21. Areas of Bounded Regions (RD Sharma - Mathematics (Volume 2) Book)

## Exercise 21.1

1

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

2

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.

3

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

4

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

5

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

6

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.

7

Sketch the graph of in [0,4] and determine the area of the region enclosed by the curve, the x - axis and the lines x = 0, x = 4

8

Find the area under the curve above x - axis from x = 0 to x = 2. Draw a sketch of curve also.

9

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

10

Draw a rough sketch of the graph of the curve and evaluate the area of the region under the curve and above the x - axis

11

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

12

Draw a rough sketch of the graph of the function y=2√1–x2,x[0,1] and evaluate the are enclosed between the curve and the x–axis.

13

Determine the area under the included between the lines x = 0 and x = 1

14

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

15

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

16

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.

17

Sketch the graph y = |x - 5|. Evaluate . What does this value of the integral represent on the graph?

18

Sketch the graph y = |x + 3|. Evaluate . What does this integral represent on the graph?

19

Sketch the graph y = |x + 1|. Evaluate . What does the value of this integral represent on the graph?

20

Draw a rough sketch of the curve xy –3x – 2y – 10 = 0, x - axis and the lines x = 3, x = 4.

21

Draw a rough sketch of the curve and find the area between x - axis, the curve and the ordinates x = 0, x = π.

22

Draw a rough sketch of the curve and find the area between x - axis, the curve and the ordinates x = 0, x = π.

23

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

24

Show that the areas under the curves y = sin x and y = sin 2x between x = 0 and are in the ration 2:3.

25

Compare the areas under the curves y = cos2x and y = sin2 x between x = 0 and x = π.

26

Find the area bounded by the ellipse and the ordinates x = ae and x = 0, where b2 = a2 (1 - e2) and e<1.

27

Find the area of the minor segment of the circle x2 + y2 = a2 cut off by the line .

28

Find the area of the region bounded by the curve x = at2, y = 2at between the ordinates corresponding t = 1 and t = 2

29

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

## Exercise 21.2

1

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.

2

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

3

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

4

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

5

Find the area of the region bounded by the curve ay2 = x3, they y-axis and the lines y = a and y = 2a.

## Exercise 21.3

1

Calculate the area of the region bounded by the parabolas y2 = 6x and x2 =6y.

2

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

3

Find the area of the region bounded by y = √x and y = x

4

Find the area bounded by the curve y = 4 – x2 and the lined y = 0, y = 3.

5

Find the area of the region .

6

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

7

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.

8

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

9

Find the area of the region {(x, y) : y28x, x2 + y2 9}

10

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

11

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

12

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

13

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.

14

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

15

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.

16

Find the area included between the parabolasy2 = 4ax and x2 = 4by.

17

Prove that the area in the first quadrant enclosed by the axis, the line x = √3y and the circle x2 + y2 = 4 is π/3.

18

Find the area of the region bounded by by y = √x and x = 2y + 3 in the first quadrant and x - axis.

19

Find the area common to the circle x2 + y2 = 16 a2 and the parabola y2 = 6ax.

OR

Find the area of the region {(x,y):y2 6ax} and {(x,y):x2 + y2 16a2}.

20

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

21

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

22

Prove that the area common to the two parabolas y = 2x2 and y = x2 + 4 is 32/3 sq. Units.

23

Using integration, find the area of the region bounded by the triangle whose vertices are

(i) ( – 1, 2), (1, 5) and (3, 4)

(ii) ( – 2, 1), (0, 4) and (2, 3)

24

Find the area of the region bounded by y = √x and y = x.

25

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

26

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

27

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

28

Find the area enclosed by the curve y = – x2 and the straight line x + y + 2 = 0

29

Find the area enclosed by the curve Y = 2 – x2 and the straight line x + y = 0.

30

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

31

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

32

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

33

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

34

Using integration find the area of the region {(x,y)|x – 1| y √5 – x2}.

35

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

36

Find the area of the region bounded by y = x and circle x2 + y2 = 32 in the 1st quadrant.

37

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

38

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

39

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.

40

Find the area of the region bounded by the curve y = √1 – x2, line y = x and the positive x - axis.

41

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

42

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

43

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

44

Using integration, find the area of the following region .

45

Using integration find the area of the region bounded by the curve , x2 + y2 – 4x = 0 and the x-axis.

46

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

47

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

48

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

49

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

50

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

51

If the area bounded by the parabola y2 = 4ax and the line y = mx is a2/12 sq. Units, then using integration, find the value of m.

52

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

## Exercise 21.4

1

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

2

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

3

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.

4

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

## MCQ

1

If the area above the x-axis, bounded by the curves y = 2kx and x = 0, and x = 2 is then the value of k is

2

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

3

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

4

The area bounded by y = 2 – x2 and x + y = 0 is

5

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

6

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

7

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

8

The area enclosed between the curves y =loge (x + e), x = loge and the x-axis is

1

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

10

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

11

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

12

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

13

The area bounded by the parabola y2 = 4ax, latus rectum and x-axis is

14

The area of the region is

15

The area common to the parabola y = 2x2 and y = x2 + 4 is

16

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

17

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

18

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

19

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

20

The area bounded by the curve y = 4x – x2 and the x-axis is

21

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

22

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

23

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

24

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

25

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

26

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

27

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

28

The area bounded by the y-axis, y = cos x and y = sin x when 0 ≤ x ≤ π/2 is

29

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

30

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

31

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