CHAPTER 8 Application of Integrals Page 299
CHAPTER 8
App lication of Integrals
OBJECTIVE QUESTION of the region AOB of the parabola y = x2 is equal to
1. The area of enclosed by y = 3x − 5 , y = 0 , x = 3 and
x = 5 is
(a) 12 sq units (b) 13 sq units
1
(c) 13 sq units (d) 14 sq units
2
Sol : OD 2017
2. The area bounded by y = log x , x -axis and ordinates
x = 1, x = 2 is
(a) 1 (log 2) 2 (b) log (2/e)
2
(c) log (4/e) (d) log 4
Sol : Delhi 2018 (a) 3 (b) 3
5 4
(c) 7 (d) 5
8 6
3. The area of the region bounded by the lines y = mx, Sol : Comp 2016, OD 2012
x = 1, x = 2 and x -axis is 6 sq units, then m is equal
to
(a) 3 (b) 1 7. The area bounded by y = sin x , x -axis and the lines
(c) 2 (d) 4 x = π is
Sol : Delhi 2015 (a) 2 sq units (b) 3 sq units
(c) 4 sq units (d) None of these
Sol : OD 2010
4. Area of a curve xy = 4 , bounded by the lines x = 1
and x = 3 and x -axis will be
(a) log 12 (b) log 64 8. The area bounded by the curve y = 1 x2 , the x -axis
2
(c) log 81 (d) log 27 and the ordinate x = 2 is
(a) 1 sq unit (b) 2 sq unit
Sol : Comp 2014
3 3
(c) 1 sq unit (d) 4 sq unit
3
Sol : Delhi 2014, OD 2010
5. The area of enclosed by y = 3x − 5 , y = 0 , x = 3 and
x = 5 is
9. Area of the region satisfying x # 2 , y # x and
(a) 12 sq units (b) 13 sq units
x $ 0 is
(c) 13 12 sq units (d) 14 sq units (a) 4 sq units (b) 1 sq unit
Sol : Foreign 2007
(c) 2 sq units (d) None of these
Sol : Foreign 2006
6. The given figure shows a TAOB and the parabola
y = x2 . The ratio of the area of the TAOB to the area
Page 300 Application of Integrals CHAPTER 8
10. The area bounded by the parabola y2 = 8x and its 19. Find the area of the region bounded by the curve
latusrectum is y = 16 − x2 and x -axis.
Sol : Comp 2016, Delhi 2013
(a) 16/3 sq units
(b) 32/3 sq units
(c) 8/3 sq units 20. Find the area of the region bounded by the curve
y = x − 2 , , x = 1, x = 3 and x -axis.
(d) 64/3 sq units
Sol : OD 2010
Sol : Delhi 2018
11. The area bounded by y = log x , x -axis and ordinates
21. Find the area bounded by y = − x2 + 2x + 3 and y = 0.
x = 1, x = 2 is Sol : SQP 2019, OD 2017
(a) 12 (log 2) 2 (b) log e2
(c) log e4 (d) log 4
Sol : Delhi 2008
SHORT ANSWER QUESTION
22. Find the area of the region bounded by the parabola
12. Sketch the region bounded by the lines 2x + y = 8,
y2 = 4ax, its axis and two ordinates x = 4 and x = 9
y = 2, y = 4 and the y -axis. Hence, obtain its area
in first quadrant.
using integration. Sol : Comp 2008
Sol : OD 2023
23. Find the area bounded by the line y = x, the x -axis
13. Using integration, find the area of the region bounded
and the lines x =− 1 and x = 2 .
by y = mx ^x > 0h, x = 1 and x = 2 and the x -axis. Sol : OD 2009
Sol : OD 2023
24. Sketch the region y = 4 − x2 and x -axis. Find the
14. Using integration, find the area of the region bounded
area of the region using integration.
by the curves y = x + 1 + 1, x =− 3, x = 3 and Sol : Foreign 2016
y = 0.
Sol : Delhi 2014
LONG ANSWER QUESTION
15. Find the area lying in the first quadrant and bounded
by the circle x2 + y2 = 4 and the lines x = 0 and x = 2.
Sol : Comp 2015, OD 2013
25. Find the area bounded by he curve x = 4 − y2 and
the y -axis.
16. Find the area of the region bounded by the curve Sol : Delhi 2017
y2 = 4x, y -axis and the line y = 3.
Sol : Foreign 2010
26. Find the area of region bounded by the curve y2 = 4x
and the line x = 4 .
17. Find the area of the region bounded by the curve Sol : OD 2017, Delhi 2014
y = cos x between x = 0 and x = π.
Sol : Foreign 2017
27. Using integration, find the area of the region bounded
18. If we draw a rough sketch of the curve y = x − 1 by the triangle whose vertices are ^- 1, 0h , ^1, 3h and
in the interval [1,5] then find under the curve and ^3, 2h .
Sol : OD 2009
between the lines x = 1 and x = 5 .
Sol : Delhi 2012
CHAPTER 8 Application of Integrals Page 301
28. Using integration, find the area of the triangular
region whose have the equation y = 2x + 1, y = 3x + 1
and x = 4 .
Sol : OD 2011
29. Using integration, find the area of TABC , the
coordinates of whose vertices are A (2, 5), B (4, 7) and
C (6, 2).
Sol : Delhi 2019, OD 2011, Comp 2010
30. Sketch the graph of y = x + 3 and evaluate the area
under the curve y = x + 3 above x -axis and between
x =− 6 to x = 0 .
Sol : OD 2011
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