RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 33 of Exercise 21.3

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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).

To find area of ΔABC with A (4,1), B(6,6) and C(8,4)

Equation of AB,





...(i)


Equation of BC,



y – 6 = – 1(x – 6)


y = – x + 12 ...(ii)


Equation of AC,



These are shown in the graph below: -



Clearly, Area of ΔABC = Area ADB + Area BDC


Area of ADB,





Similarly,








Thus, Area ABC = Area ADB + Area BDC


=


= 64 – 56


= 7 sq. units


The area of the triangle ABC coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4) is 7 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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