RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 6 of Exercise 21.3

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6

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

Here we have to find the area of the triangle whose triangle are A(2,1), B(3,4) and C(5,2) as shown below.



The equation of AB,





Y = 3x – 5 ...(i)


The equation of BC,




...(ii)


The equation of AC,






Now the required area (A) =


[(Area between line AB and x - axis) - (Area between line AC and x - axis) from x = 2 to x = 3]


+ [(Area between line BC and x - axis) - (Area between line AC and x - axis) from x = 3 to x = 5]











The area of the region bounded by the triangle whose vertices are (2,1), (3,4) and (5,2) is 4 sq. units


Chapter Exercises

Exercise 21.2
Exercise 21.3
Exercise 21.4
Exercise 21.1
MCQ

More Exercise Questions

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) : y2≤8x, 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.

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