Application Of Integrals

01

PYQ 2021

medium

mathematics ID: cuet-ug-

The area of the region bounded by the line and the curve is ________.

1

square units

2

0 square unit

3

1 square unit

4

32 square units

02

PYQ 2024

medium

mathematics ID: cuet-ug-

The value of, where [ ] denotes the greatest integer function, is:

1

2

3

4

03

PYQ 2024

medium

mathematics ID: cuet-ug-

The integral of the function is:

1

, where is an arbitrary constant

2

, where is an arbitrary constant

3

, where is an arbitrary constant

4

, where is an arbitrary constant

04

PYQ 2024

medium

mathematics ID: cuet-ug-

For if the point satisfies, then the constant of integration of the given integral is:

1

2

3

4

0

05

PYQ 2024

medium

mathematics ID: cuet-ug-

The value of is:

1

2

3

4

06

PYQ 2024

hard

mathematics ID: cuet-ug-

The value of is:

1

2

3

4

07

PYQ 2025

hard

mathematics ID: cuet-ug-

The area (in sq. units) of the region bounded by the parabola y2 = 4x and the line x = 1 is

1

2

3

4

08

PYQ 2025

medium

mathematics ID: cuet-ug-

$The area (in sq. units) of the region bounded by y =, x [0,1] and x-axis is equal to$

1

$1$

2

$2$

3

$(D)$

4

Official Solution

Correct Option: (3)

$Step 1: Understanding the Concept: The given curve represents a portion of an ellipse. We can find the area either by setting up a definite integral or by using the geometric formula for the area of an ellipse. Step 2: Key Formula or Approach: Method 1: Geometric Interpretation Rearrange the equation of the curve: Square both sides (assuming y 0): This is the standard equation of an ellipse centered at the origin with semi-minor axis a = 1 and semi-major axis b = 2. The area of a full ellipse is . The given constraints are (from the positive square root) and, which means we are finding the area of the ellipse in the first quadrant. Method 2: Integration The area is given by the integral . Step 3: Detailed Explanation: Using Method 1 (Geometry): The total area of the ellipse is . The required area is the portion in the first quadrant (where x is from 0 to 1 and y is positive). This is exactly one-quarter of the total area of the ellipse. Using Method 2 (Integration): We use the standard integral formula . Here, a = 1. Step 4: Final Answer: The area of the region is} sq. units.$

09

PYQ 2025

medium

mathematics ID: cuet-ug-

$The area (in sq. units) of the region bounded by the curve, the x-axis and the ordinates x = -1 and x = 1 is equal to$

1

2

$1$

3

$(D)$

4

Official Solution

Correct Option: (3)

$Note: The provided question with yields an answer of, which is not among the options. It is highly likely that the question intended to ask for, as handwritten '3' and '5' can look similar, and the correct answer for is one of the options. We will proceed with this assumption. Step 1: Understanding the Concept: We need to find the area bounded by a curve and the x-axis. Since area is always a non-negative quantity, we must integrate the absolute value of the function, . The function is negative for and positive for . Step 2: Key Formula or Approach: The area is given by the definite integral of from -1 to 1: We must split the integral at where the function's sign changes. Step 3: Detailed Explanation: Let's split the integral into two parts: For, is negative, so . For, is positive, so . Now, we integrate: Evaluate the first part: Evaluate the second part: The total area is the sum of the two parts: Step 4: Final Answer: Assuming the intended curve was, the area is square units.$

10

PYQ 2025

medium

mathematics ID: cuet-ug-

$The area (in sq. units) of the region bounded by the parabola y2 = 4x and the line x = 1 is$

1

2

3

4

Official Solution

Correct Option: (4)

$Step 1: Understanding the Concept: We need to find the area of a region enclosed by a curve and a line. This is a classic application of definite integration. The curve is a parabola opening to the right with its vertex at the origin. Step 2: Key Formula or Approach: The area of a region bounded by a curve, the x-axis, and the lines and is given by . Since the parabola is symmetric about the x-axis, we can find the area in the first quadrant and multiply it by 2. Step 3: Detailed Explanation: The given parabola is . For the upper half of the parabola (in the first quadrant), we have . The region is bounded by (the y-axis, where the parabola starts) and . The area in the first quadrant is given by the integral: Using the power rule for integration: This is the area of the region above the x-axis. Due to symmetry, the area of the region below the x-axis is also . The total area is: Step 4: Final Answer: The total area of the region is} sq. units.$

11

PYQ 2025

medium

mathematics ID: cuet-ug-

$The area (in sq. units) of the region bounded by the curve, the x-axis and the ordinates x = -1 and x = 1 is equal to$

1

2

$1$

3

$(D)$

4

Official Solution

Correct Option: (3)

$Note: The provided question with yields an answer of, which is not among the options. It is highly likely that the question intended to ask for, as handwritten '3' and '5' can look similar, and the correct answer for is one of the options. We will proceed with this assumption. Step 1: Understanding the Concept: We need to find the area bounded by a curve and the x-axis. Since area is always a non-negative quantity, we must integrate the absolute value of the function, . The function is negative for and positive for . Step 2: Key Formula or Approach: The area is given by the definite integral of from -1 to 1: We must split the integral at where the function's sign changes. Step 3: Detailed Explanation: Let's split the integral into two parts: For, is negative, so . For, is positive, so . Now, we integrate: Evaluate the first part: Evaluate the second part: The total area is the sum of the two parts: Step 4: Final Answer: Assuming the intended curve was, the area is square units.$

12

PYQ 2025

medium

mathematics ID: cuet-ug-

$The area (in sq. units) of the region bounded by y =, x [0,1] and x-axis is equal to$

1

$1$

2

$2$

3

$(D)$

4

Official Solution

Correct Option: (3)

$Step 1: Understanding the Concept: The given curve represents a portion of an ellipse. We can find the area either by setting up a definite integral or by using the geometric formula for the area of an ellipse. Step 2: Key Formula or Approach: Method 1: Geometric Interpretation Rearrange the equation of the curve: Square both sides (assuming y 0): This is the standard equation of an ellipse centered at the origin with semi-minor axis a = 1 and semi-major axis b = 2. The area of a full ellipse is . The given constraints are (from the positive square root) and, which means we are finding the area of the ellipse in the first quadrant. Method 2: Integration The area is given by the integral . Step 3: Detailed Explanation: Using Method 1 (Geometry): The total area of the ellipse is . The required area is the portion in the first quadrant (where x is from 0 to 1 and y is positive). This is exactly one-quarter of the total area of the ellipse. Using Method 2 (Integration): We use the standard integral formula . Here, a = 1. Step 4: Final Answer: The area of the region is} sq. units.$

High-Yield Trend

Chapter Questions
12 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Application Of Integrals

High-Yield Trend

Chapter Questions 12 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Chapter Questions
12 MCQs