![CHAPTER – 8
APPLICATION OF INTEGRATION
6 Marks Questions:
1. Find the area of the region bounded by y2
= 9x , x = 2, x =4 and the x axis is in first quadrant.
[ 16 - 4 2 ]
2. Find the area bounded by curve x2
= 4y and the line x = 4y – 2.
3. Find the area of the region bounded by two parabolas y2
= x and x2
=y. [ 1/3]
4. Find the area of the region between parabola x2
= 4y and y2
= 4x . [ 16/3]
5. Find the area bounded by curves x2
+ y2
= 1 and (x – 1)2
+ y2
=1 .
2𝜋
3
−
3
2
6. Find the area of the region included between the curve 4y = 3x2
and line 2y = 3x + 12 . [27]
7. Using integration, find the area lying above x – axis and includes between x2
+ y2
= 8x and line
2y = 3x + 12. [27]
8. Find the area of that part of the circle x2
+ y2
= 16 which is exterior to parabola y2
= 6x.
9. Using integration, find the area between x2
+ y2
= 4 and (x – 2)2
+ y2
= 4.
8𝜋
3
+ 2 3 .
10. Using integration find the area of region bounded by the ∆ whose vertices are
(1,0) (2,2) and (3,1) . [3/2]
11. Find the area of region {(x,y) : x2
≤ 𝑦 ≤ 𝑥 } .
12. Find the area of region {(x,y) : y2
≤ 4𝑥 , 4𝑥2
+ 4𝑦2
≤ 9} .
13. Prove that the area between two parabolas y2
4ax and x2
= 4ay is 16 a2
/ 3 sq units.
14. Sketch the graph of y = 𝑥 + 3 and Evaluate 𝑥 + 3
0
−6
dx .
15. Find the area enclosed by the circle x2
+ y2
= 4 and line x + y = 2 .
16. Find the area of smaller region bounded by the ellipse
𝑥2
𝑎2 +
𝑦2
𝑏2 = 1 and line
𝑥
𝑎
+
𝑦
𝑏
= 1 .
17. Draw and sketch the following region and find its area {(x , y) : 𝑥2
+ 𝑦2
≤ 1 ≤ 𝑥 + 𝑦} .
18. Using integration , find the area of ∆ ABC where A (2,3), B (4,7), C (6,2).
19. Find the area bounded by curve y = cos x between x = 0 and x = 2𝜋 .
20. Find the area of the region : {(x,y) : 0 ≤ y ≤ 𝑥2
, 0 ≤ 𝑦 ≤ 𝑥 + 2, 0 ≤ 𝑥 ≤ 3} .[43/6]
21. Using the method of integration, find the area of the region bounded by the lines
2x+y = 4, 3x – 2y = 6, x – 3y +5 = 0. [7/2]
22. Find the area of the region included between y2
= x and line x + y = 2. [9/2]
23. Find the area between the curves y – x2
and y = x .](https://image.slidesharecdn.com/class12chapter8forblog-170729083243/85/Class-12-chapter-8-1-320.jpg)
![24. Using the method of integration, find the area of the ∆ ABC, coordinate of whose vertices are A
(2,0), B (4,5), C (6,3). [7]
25. Find the area bounded by the circle x2
+ y2
= 16 and the line y = x in the first Quadrant.[2𝜋]
26. Using integration, find the area of the following region. 𝑥, 𝑦 :
𝑥2
9
+
𝑦2
4
≤ 1 ≤
𝑥
3
+
𝑦
2
.
27. Find the area of the region {(x, y) : x2
+ y2
≤ 4, 𝑥 + 𝑦 ≥ 2}.
28. Find the area lying above x – axis and included between the circle x2
+ y2
= 8x and the
parabola y2
= 4x.
29. Find the area of that part of the circle x2
+ y2
= 16 which is exterior to the parabola y2
= 6x.
30. Find the area of the region enclosed between the two circles x2
+ y2
= 9 and (x – 3)2
+ y2
= 9.
31. Using integration, find the area of the following region : {(x, y) : 𝑥 − 1 ≤ 𝑦 ≤ 5 − 𝑥2 } .
32. Using integration find the area of the triangular region whose sides have equations :
Y = 2x + 1, y = 3x + 1 and x = 4.](https://image.slidesharecdn.com/class12chapter8forblog-170729083243/85/Class-12-chapter-8-2-320.jpg)
Class 12 chapter 8
- 1. CHAPTER – 8 APPLICATION OF INTEGRATION 6 Marks Questions: 1. Find the area of the region bounded by y2 = 9x , x = 2, x =4 and the x axis is in first quadrant. [ 16 - 4 2 ] 2. Find the area bounded by curve x2 = 4y and the line x = 4y – 2. 3. Find the area of the region bounded by two parabolas y2 = x and x2 =y. [ 1/3] 4. Find the area of the region between parabola x2 = 4y and y2 = 4x . [ 16/3] 5. Find the area bounded by curves x2 + y2 = 1 and (x – 1)2 + y2 =1 . 2𝜋 3 − 3 2 6. Find the area of the region included between the curve 4y = 3x2 and line 2y = 3x + 12 . [27] 7. Using integration, find the area lying above x – axis and includes between x2 + y2 = 8x and line 2y = 3x + 12. [27] 8. Find the area of that part of the circle x2 + y2 = 16 which is exterior to parabola y2 = 6x. 9. Using integration, find the area between x2 + y2 = 4 and (x – 2)2 + y2 = 4. 8𝜋 3 + 2 3 . 10. Using integration find the area of region bounded by the ∆ whose vertices are (1,0) (2,2) and (3,1) . [3/2] 11. Find the area of region {(x,y) : x2 ≤ 𝑦 ≤ 𝑥 } . 12. Find the area of region {(x,y) : y2 ≤ 4𝑥 , 4𝑥2 + 4𝑦2 ≤ 9} . 13. Prove that the area between two parabolas y2 4ax and x2 = 4ay is 16 a2 / 3 sq units. 14. Sketch the graph of y = 𝑥 + 3 and Evaluate 𝑥 + 3 0 −6 dx . 15. Find the area enclosed by the circle x2 + y2 = 4 and line x + y = 2 . 16. Find the area of smaller region bounded by the ellipse 𝑥2 𝑎2 + 𝑦2 𝑏2 = 1 and line 𝑥 𝑎 + 𝑦 𝑏 = 1 . 17. Draw and sketch the following region and find its area {(x , y) : 𝑥2 + 𝑦2 ≤ 1 ≤ 𝑥 + 𝑦} . 18. Using integration , find the area of ∆ ABC where A (2,3), B (4,7), C (6,2). 19. Find the area bounded by curve y = cos x between x = 0 and x = 2𝜋 . 20. Find the area of the region : {(x,y) : 0 ≤ y ≤ 𝑥2 , 0 ≤ 𝑦 ≤ 𝑥 + 2, 0 ≤ 𝑥 ≤ 3} .[43/6] 21. Using the method of integration, find the area of the region bounded by the lines 2x+y = 4, 3x – 2y = 6, x – 3y +5 = 0. [7/2] 22. Find the area of the region included between y2 = x and line x + y = 2. [9/2] 23. Find the area between the curves y – x2 and y = x .
- 2. 24. Using the method of integration, find the area of the ∆ ABC, coordinate of whose vertices are A (2,0), B (4,5), C (6,3). [7] 25. Find the area bounded by the circle x2 + y2 = 16 and the line y = x in the first Quadrant.[2𝜋] 26. Using integration, find the area of the following region. 𝑥, 𝑦 : 𝑥2 9 + 𝑦2 4 ≤ 1 ≤ 𝑥 3 + 𝑦 2 . 27. Find the area of the region {(x, y) : x2 + y2 ≤ 4, 𝑥 + 𝑦 ≥ 2}. 28. Find the area lying above x – axis and included between the circle x2 + y2 = 8x and the parabola y2 = 4x. 29. Find the area of that part of the circle x2 + y2 = 16 which is exterior to the parabola y2 = 6x. 30. Find the area of the region enclosed between the two circles x2 + y2 = 9 and (x – 3)2 + y2 = 9. 31. Using integration, find the area of the following region : {(x, y) : 𝑥 − 1 ≤ 𝑦 ≤ 5 − 𝑥2 } . 32. Using integration find the area of the triangular region whose sides have equations : Y = 2x + 1, y = 3x + 1 and x = 4.