VBQs Application of Integrals Class 12 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 12 Mathematics with solutions. The following Application of Integrals Class 12 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 12 examinations.

**Application of Integrals VBQs Class 12 Mathematics**

**Question. If the area (in sq. units) of the region {(x, y) : y ^{2} ≤ 4x, x + y ≤ 1, x ≥ 0, y ≥ 0} is a √2 + b, then a – b is equal to :**

(a) 10/3

(b) 6

(c) 8/3

(d) − 2/3

## Answer

B

**Question. If the area (in sq. units) bounded by the parabola y ^{2} = 4λx and the line y = λx, λ > 0, is 1/9, then λ is equal to :**

(a) 2√6

(b) 48

(c) 24

(d) 4√3

## Answer

C

**Question. If the area of the region bounded by the curves, y = x ^{2}, y = 1/x and the lines y = 0 and x = t (t > 1) is 1 sq. unit, then t is equal to**

(a) 4/3

(b) e

^{2/3}

(c) 3/2

(d) e

^{3/2}

## Answer

B

**Question. The area (in sq. units) of the region {x∈ R : x ≥ 0, y ≥ 0, y ≥ x – 2 and y ≤ √x}, is**

(a) 13/3**(b)** **10/3**

(c) 5/3

(d) 8/3

## Answer

B

**Question. The area of the region A = {(x, y): 0 ≤ y ≤ x |x| + 1 and – 1 ≤ x ≤ 1} in sq. units is:**

(a) 2/3

(b) 2

(c) 4/3

(d) 1/3

## Answer

B

**Question. The parabolas y ^{2} = 4x and x^{2} = 4y divide the square region bounded by the lines x = 4, y = 4 and the coordinate axes. If S1 , S2 , S3 are respectively the areas of these parts numbered from top to bottom; then S_{1} : S_{2} : S_{3} is**

(a) 1 : 2 : 1

(b) 1 : 2 : 3

(c) 2 : 1 : 2

(d) 1 : 1 : 1

## Answer

D

**Question. The area of the region bounded by the curves y =| x – 2 |, x = 1, x = 3 and the x-axis is**

(a) 4

(b) 2

(c) 3

(d) 1

## Answer

D

**Question. Let ƒ(x) be a non – negative continuous function such that the area bounded by the curve y = ƒ(x), x – axis and the ordinates x = π/4 and x = b > π/4 is **

**Then ƒ(π/2) is**

## Answer

D

**Question. The area of the region above the x-axis bounded by the curve y = tan x, 0 ≤ x ≤ π/2 and the tangent to the curve at x = π/4 is :**

## Answer

A

**Question. Let A = {(x, y): y ^{2} ≤ 4x, y – 2x ≥ – 4}. The area (in square units) of the region A is:**

(a) 8

(b) 9

(c) 10

(d) 11

## Answer

B

**Question. The area (in sq. units) of the region A = {(x, y) : |x| + |y| ≤ 1, 2y ^{2} ≥ |x|} is :**

(a) 1/3

(b) 7/6

(c) 1/6

(d) 5/6

## Answer

D

**Question. The area (in sq. units) of the region A = {(x, y) : (x -1)[x] ≤ y ≤ 2√x, 0 ≤ x ≤ 2}, where [t] denotes the greatest integer function, is :**

## Answer

A

**Question. The area (in sq. units) of the region {(x, y) : 0 ≤ y ≤ x ^{2} + 1, 0 ≤ y ≤ x + 1, 1/2 ≤ x ≤ 2} is :**

(a) 23/16

(b) 79/24

(c) 79/16

(d) 23/6

## Answer

B

**Question. The area of the region, enclosed by the circle x ^{2} + y^{2} = 2 which is not common to the region bounded by the parabola y^{2} = x and the straight line y = x, is:**

(a) (24π – 1)

(b) (6π – 1)

(c) (12π – 1)

(d) (12π – 1)/6

## Answer

D

**Question. The area (in sq. units) of the region {(x, y) ∈ R ^{2}: x^{2} ≤ y ≤ |3 – 2x|, is:**

(a) 32/3

(b) 34/3

(c) 29/3

(d) 31/3

## Answer

A

**Question. The area bounded by the parabola y ^{2} = 4x and the line 2x – 3y + 4 = 0, in square unit, is**

(a) 2/5

(b) 1/3

(c) 1

(d) 1/2

## Answer

B

**Question. The area of the region bounded by the curve y = x ^{3}, and the lines, y = 8, and x = 0, is**

(a) 8

(b) 12

(c) 10

(d) 16

## Answer

B

**Question. The region represented by |x – y| ≤ 2 and |x + y| ≤ 2 is bounded by a :**

(a) square of side length 2√2 units

(b) rhombus of side length 2 units

(c) square of area 16 sq. units

(d) rhombus of area 8√2 sq. units

## Answer

A

**Question. The area (in sq. units) of the smaller portion enclosed between the curves, x ^{2} + y^{2} = 4 and y^{2} = 3x, is :**

(a) 1/(2√3) + π/3

(b) 1/√3 + 2π/3

(c) 1/(2√3) + 2π/3

(d) 1/√3 + 4π/3

## Answer

D

**Question. The area (in sq. units) of the region {(x, y) : y ^{2} ≥ 2x and x^{2} + y^{2} ≤ 4x, x ≥ 0, y ≥ 0} is :**

(a) π − (4√2)/3

(b) π/2 − (2√2)/3

(c) π − 4/3

(d) π − 8/3

## Answer

D

**Question. The area (in sq. units) of the region A = {(x, y) : y ^{2}/2 ≤ x ≤ y + 4} is :**

(a) 53/3

(b) 30

(c) 16

(d) 18

## Answer

D

**Question. The area (in sq. units) of the region bounded by the curve x ^{2} = 4y and the straight line x = 4y – 2 is :**

(a) 5/4

(b) 9/8

(c) 7/8

(d) 3/4

## Answer

B

**Question. The area (in sq. units) in the first quadrant bounded by the parabola, y = x ^{2} + 1, the tangent to it at the point (2, 5) and the coordinate axes is :**

(a) 8/3

(b) 37/24

(c) 187/24

(d) 14/3

## Answer

B

**Question. The area of the region bounded by the parabola (y – 2) ^{2} = x –1, the tangent of the parabola at the point (2, 3) and the x-axis is:**

(a) 6

(b) 9

(c) 12

(d) 3

## Answer

B

**Question. The area of the plane region bounded by the curves x + 2y ^{2} = 0 and x + 3y^{2} = 1is equal to**

(a) 5/3

(b) 1/3

(c) 2/3

(d) 4/3

## Answer

D

**Question. The area (in sq. units) of the region {(x, y) ∈R ^{2}|4x^{2} ≤ y ≤ 8x + 12} is:**

(a) 125/3

(b) 128/3

(c) 124/3

(d) 127/3

## Answer

B

**Question. For a > 0, let the curves C _{1}: y^{2} = ax and C_{2}: x^{2}= ay intersect at origin O and a point P. Let the line x = b (0 < b < a) intersect the chord OP and the x-axis at points Q and R, respectively. If the line x = b bisects the area bounded by the curves, C_{1} and C_{2}, and the area of ΔOQR = 1/2, then ‘a’ satisfies the equation:**

(a) x

^{6}– 6x

^{3}+ 4 = 0

(b) x

^{6}– 12x

^{3}+ 4 = 0

(c) x

^{6}+ 6x

^{3}– 4 = 0

(d) x

^{6}– 12x

^{3}– 4 = 0

## Answer

B

**Question. The area enclosed between the curves y ^{2} = x and y = | x | is**

(a) 1/6

(b) 1/3

(c) 2/3

(d) 1

## Answer

A

**Question. If the area enclosed between the curves y = kx ^{2} and x = ky^{2}, (k > 0), is 1 square unit. Then k is:**

(a) √3/2

(b) 1/√3

(c) √3

(d) 2/√3

## Answer

B

**Question. The area (in sq. units) bounded by the parabola y = x ^{2} –1, the tangent at the point (2, 3) to it and the y-axis is:**

(a) 8/3

(b) 32/3

(c) 56/3

(d) 14/3

## Answer

A

**Question. The area (in sq. units) of the region {(x, y) : x ≥ 0, x + y ≤ 3, x ^{2} ≤ 4y and y ≤ 1 + √x } is :**

(a) 5/2

(b) 59/12

(c) 3/2

(d) 7/3

## Answer

A

**Question. The area (in sq. units) of the region enclosed by the curves y = x ^{2} – 1 and y = 1 – x^{2} is equal to:**

(a) 4/3

(b) 8/3

(c) 7/2

(d) 16/3

## Answer

B

**Question. Consider a region R ={(x, y) ∈ R ^{2} : x^{2} ≤ y ≤ 2x}. If a line y = a divides the area of region R into two equal parts, then which of the following is true?**

(a) a

^{3}− 6a

^{2}+ 16 = 0

(b) 3a

^{2}− 8a

^{3/2}+ 8 = 0

(c) 3a

^{2}− 8a + 8 = 0

(d) a

^{3}− 6a

^{3/2}− 16 = 0

## Answer

B

**Question. The area (in sq. units) of the region described by A = {(x, y)| y ≥ x ^{2} – 5x + 4, x + y ≥ 1, y ≤ 0} is:**

(a) 19/6

(b) 17/6

(c) 7/2

(d) 13/6

## Answer

A

**Question. The area (in sq. units) of the region described by {(x, y) : y ^{2} ≤ 2x and y ≥ 4x – 1} is**

(a) 15/64

(b) 9/32

(c) 7/32

(d) 5/64

## Answer

B

**Question. The area of the region bounded by the curves y = |x – 1| and y = 3 – |x| is**

(a) 6 sq. units

(b) 2 sq. units

(c) 3 sq. units**(d) 4 sq. units**

## Answer

D

**Question. The area bounded by the curves y = lnx, y = ln |x|,y = | ln x | and y = | ln |x| | is**

(a) 4sq. units

(b) 6 sq. units

(c) 10 sq. units

(d) none of these

## Answer

A

**Question. The area (in square units) of the region bounded by the curves y + 2x ^{2} = 0 and y + 3x^{2} = 1, is equal to :**

(a) 3/5

(b) 1/3

(c) 4/3

(d) 3/4

## Answer

C

**Question. The area of the region described by A = {(x,y) : x ^{2} + y^{2} ≤ 1 and y^{2} ≤ 1 − x} is :**

(a) π/2 − 2/3

(b) π/2 + 2/3

(c) π/2 + 4/3

(d) π/2 − 4/3

## Answer

C

**Question. The area (in square units) bounded by the curves y = √x , 2y – x + 3 = 0, x-axis, and lying in the first quadrant is :**

(a) 9

(b) 36

(c) 18

(d) 27/4

## Answer

A

**Question. The area under the curve y = | cos x – sin x |, 0 ≤ x ≤ π/2, and above x-axis is :**

(a) 2√2

(b) 2√2 – 2

(c) 2√2 + 2

(d) 0

## Answer

B

**Question. Given: **

**and g(x) = (x − 1/2) ^{2}, x ∈ R. Then the area (in sq. units) of the region bounded by the curves, y = ƒ(x) and y = g(x) between the lines, 2x = 1 and 2x = √3, is :**

(a) 1/3 + √3/4

(b) √3/4 − 1/3

(c) 1/2 − √3/4

(d) 1/2 + √3/4

## Answer

B

**Question. The area (in sq. units) of the region A = {(x, y) ∈R × R|0 d” x d”3, 0 d” y d” 4, y d” x ^{2} + 3x} is :**

(a) 53/6

(b) 8

(c) 59/6

(d) 26/3

## Answer

C

**Question. The area of the region (in sq. units), in the first quadrant bounded by the parabola y = 9x ^{2} and the lines x = 0, y = 1 and y = 4, is :**

(a) 7/9

(b) 14/3

(c) 7/3

(d) 14/9

## Answer

D

**Question. The area bounded by the curve y = ln (x) and the lines y = 0, y = ln (c) and x = 0 is equal to :**

(a) 3

(b) 3 ln (c) – 2

(c) 3 ln (c) + 2

(d) 2

## Answer

D

**Question. The area (in sq. units) of the region bounded by the curves y = 2 ^{x} and y = |x + 1|, in the first quadrant is :**

(a) log

_{e}2 + 3/2

(b) 3/2

(c) 1/2

(d) 3/2 − 1/(log

_{e}2)

## Answer

D

**Question. The area (in sq. units) of the region A = {(x, y) : x ^{2} ≤ y ≤ x + 2} is:**

(a) 10/3

(b) 9/2

(c) 31/6

(d) 13/6

## Answer

B

**Question. The area between the parabolas x ^{2} = y/4 and x^{2} = 9y and the straight line y = 2 is :**

(a) 20√2

(b) 10√2 / 3

(c) 20√2 / 3

(d) 10√2

## Answer

C

**Question. If a straight line y – x = 2 divides the region x ^{2} + y^{2} ≤ 4 into two parts, then the ratio of the area of the smaller part to the area of the greater part is**

(a) 3π – 8 : π + 8

(b) π – 3 : 3π + 3

(c) 3π – 4 : π + 4

(d) π – 2 : 3π + 2

## Answer

D

**Question. Let g(x) = cos x ^{2} , f(x) = √x , and α, β (α < β) be the roots of the quadratic equation 18x^{2} – 9πx + π^{2} = 0 . Then the area (in sq. units) bounded by the curve y = (gof)(x) and the lines x = α,x = β and y = 0 , is :**

(a) 1/2 (√3 + 1)

(b) 1/2 (√3 − √2)

(c) 1/2 (√2 − 1)

(d) 1/2 (√3 − 1)

## Answer

D

**Question. Let ƒ : [ – 2, 3] → [0, ∞ ) be a continuous function such that ƒ(1– x) =ƒ(x) for all x ∈ [-2, 3]. If R _{1} is the numerical value of the area of the region bounded by y = ƒ(x), x = –2, x = 3 and the axis of x and **

(a) 3R_{1} = 2R_{2}

(b) 2R_{1} = 3R_{2}

(c) R_{1} = R_{2}

(d) R_{1} = 2R_{2}

## Answer

D

**Question. The area enclosed by the curves y = x ^{2}, y = x^{3}, x = 0 and x = p, where p > 1, is 1/6. The p equals**

(a) 8/3

(b) 16/3

(c) 2

(d) 4/3

## Answer

D

**Question. The area of the region enclosed by the curves y = x, x = e, y = 1/x and the positive x-axis is**

(a) 1 square unit

(b) 3/2 square units

(c) 5/2 square units

(d) 1/2 square unit

## Answer

B

**Question. The area bounded by the curves y = cos x and y = sin x between the ordinates x = 0 and x = 3π/2 is**

(a) 4√2 + 2

(b) 4√2 -1

(c) 4√2 +1

(d) 4√2 – 2

## Answer

D

**Question. Let S(a) = {(x, y) : y ^{2} ≤ x, 0 ≤ x ≤ a} and A(a) is area of the region S(a). If for a λ, 0 < λ < 4, A(λ) : A(a) = 2 : 5, then λ equals :**

(a) 2(4/25)

^{1/3}

(b) 2(2/5)

^{1/3}

(c) 4(2/5)

^{1/3}

(d) 4(4/25)

^{1/3}

## Answer

D

**Question. The area (in sq. units) of the region bounded by the parabola, y = x ^{2} + 2 and the lines, y = x + 1, x = 0 and x = 3, is :**

(a) 15/4

(b) 21/2

(c) 17/4

(d) 15/2

## Answer

D

**Question. The area enclosed between the curve y = log _{e} (x + e) and the coordinate axes is**

**(a) 1**

(b) 2

(c) 3

(d) 4

## Answer

A

**Question. If y = ƒ(x) makes +ve intercept of 2 and 0 unit on x and y axes and encloses an area of 3/4 square unit with the axes then **

(a) 3/2

(b) 1

(c) 5/4

(d) –3/4

## Answer

D

**Question. The parabola y ^{2} = x divides the circle x^{2} + y^{2} = 2 into two parts whose areas are in the ratio**

(a) 9π + 2 : 3π – 2

(b) 9π – 2 : 3π + 2

(c) 7π – 2 : 2π – 3

(d) 7π + 2 : 3π + 2

## Answer

B

**Question. The area bounded by the curves y ^{2} = 4x and x^{2} = 4y is:**

(a) 32/3 sq units

(b) 16/3 sq units

(c) 8/3 sq. units

(d) 0 sq. units

## Answer

B