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QUESTION BANK mathematics (APPLICATION OF INTEGRALS)
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THIS PDF CONTAINS QUESTION BANK OF CLASS 12TH CHAPTER APPLICATION OF INTEGRALS ALONG WITH THE ANSWER KEY AT THE END
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Application of IntegralsQUICK RECAP
AREA UNDER SIMPLE CURVES Y
8 Area of the region bounded by the curve
y = f(x), x-axis and the lines x = a and
X
x = b (b > a) is,
b b
Area = ∫ y dx = ∫ f (x) dx
a a AREA BETWEEN TWO CURVES
Y 8 Area of the region between two curves
y = f(x), y = g(x) and the lines x = a, x = b is,
b
Area = ∫ [ f (x ) − g (x )] dx , f (x ) ≥ g (x ) in [a, b ]
a
X
Y
8 Area of the region bounded by the curve x = g(y),
y-axis and the lines y = a and y = b (b > a)
b b
is, Area = ∫ x dy = ∫ g ( y ) dy X
a a
Y If f (x ) ≥ g (x ) in [a, c] and f(x) ≤ g(x) in
[c, b], where a < c < b , then
g c b
8 Area = ∫ [ f (x ) − g (x )] dx + ∫ [ g (x ) − f (x )] dx
X a c
Y
y = f(x) y = g(x)
8 Area of the region bounded by the curve y = g(x) y = f(x)
y = f(x), some portion of which is above the
X
x-axis and some below the x-axis is, O x=a x=c x=b
c b
8 Area of shaded portion as shown in figure,
Area = ∫ f (x ) dx + ∫ f (x ) dx
c b
a c
Area = ∫ f (x )dx + ∫ g (x )dx
a c
Y
X
, Previous Years’ CBSE
PREVIOUS Board
YEARS MCQS Questions
8.2 Area under Simple Curves 11. Find the area of the region bounded by the
parabola y = x2 and y = |x|. (AI 2013)
LA 2 (6 marks)
12. Using integration, find the area of the region
1. Using integration, find the smaller area bounded by the curves y = x2 and y = x.
enclosed by the circle x2 + y2 = 4 and the line (Delhi 2013C)
x + y = 2. (2020) 13. Using integration, find the area of the region
2. Using integration, find the area of the region enclosed by the curves y2 = 4x and y = x.
in the first quadrant enclosed by the x-axis, (Delhi 2013C)
the line y = x and the circle x2 + y2 = 32. 14. Find the area of the region
(2018, Delhi 2014) {(x, y) : x2 + y2 ≤ 4, x + y ≥ 2}. (AI 2012)
3. Find the area bounded by the circle x2 + y2 = 16 15. Draw the graph of y = |x + 1|and using
and the line 3y = x in the first quadrant, integration, find the area below y = |x + 1|,
using integration. (Delhi 2017) above x - axis and between x = – 4 to x = 2.
(Delhi 2012C)
4. Find the area enclosed between the parabola
4y = 3x2 and the straight line 3x – 2y + 12 = 0. 16. Using integration, find the area of the region
(AI 2017, 2015C) given by {(x, y) : x2 ≤ y ≤|x|}
(AI 2012C, Delhi 2011C)
5. Using integration, find the area of the region
bounded by the line x – y + 2 = 0, the curve 17. Sketch the graph of y = | x + 3 | and evaluate
x = y and y – axis. (Foreign 2015) the area under the curve y = | x + 3|,above
x-axis and between x = – 6 to x = 0.
6. Find the area of the region in the first
(AI 2011)
quadrant enclosed by the y-axis, the line y = x
and the circle x2 + y2 = 32, using integration. 18. Find the area of the region
(Delhi 2015C) {(x, y) : x2 + y2 ≤ 1 ≤ x + y}. (AI 2011C)
7. Find the area of the smaller region bounded
8.3 Area between Two Curves
x2 y2
by the ellipse + = 1 and the line LA 2 (6 marks)
9 4
x y
+ = 1 . (Foreign 2014) 19. Using integration, find the area of the region
3 2
bounded by the triangle whose vertices are
8. Using integration, find the area of the region
(2, –2), (4, 5) and (6, 2). (2020)
bounded by the curves :
y = |x + 1| + 1, x = – 3, x = 3, y = 0 20. Using integration, find the area of the region
(Delhi 2014C) bounded by the lines
x – y = 0, 3x – y = 0 and x + y = 12. (2020)
9. Using integration, find the area bounded by
the curve x2 = 4y and the line x = 4y – 2. 21. Using integration find the area of the region
(Delhi 2014C, 2013, 2013C) bounded between the two circles x2 + y2 = 9
10. Using integration, find the area of the region and (x – 3)2 + y2 = 9. (2020)
in the first quadrant enclosed by the x – axis, 22. Using integration, find the area of triangle
the line y = x and the circle x2 + y2 = 18. ABC, whose vertices are A(2, 5), B(4, 7) and
(AI 2014C) C(6, 2). (Delhi 2019)
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