Are these practice questions for Application of Integrals based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 12 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Application of Integrals free?

Yes. Every chapter includes free practice questions for Class 12 - Science students to get you started.

What types of questions are included?

The practice set for Application of Integrals mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Application of Integrals for full coverage.

How many practice questions are there for Application of Integrals?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Application of Integrals Class 12 Mathematics Practice Questions with Answers

easy

Calculate the area bounded by the parabola

x = y^2

, the y-axis, and the lines

y=0

and

y=2

.

Try solving it in your notebook first, then check the solution.

easy

Formulate the definite integral that represents the area of the region bounded by the parabola

x = 8 + 2y - y^2

, the y-axis, and the lines

y=-1

and

y=3

. Do not evaluate the integral.

Try solving it in your notebook first, then check the solution.

easy

Calculate the area of the region bounded by the curve

y=x^2

, the x-axis, and the lines

x=0

and

x=2

.

Try solving it in your notebook first, then check the solution.

easy

Identify the definite integral that represents the area bounded by the curve

y = \cos(x)

and the x-axis between

x=0

and

x = \frac{\pi}{2}

. Do not evaluate.

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easy

Describe how the property of symmetry is used to simplify the calculation of the area of the circle

x^2 + y^2 = a^2

.

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easy

Find the area of the region bounded by the line

y=2x

, the x-axis, and the ordinates

x=1

and

x=3

.

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easy

Identify the formula for the total area of an ellipse with the equation

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

.

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easy

State the formula for the area of the region bounded by the curve

x=g(y)

, the y-axis, and the lines

y=c

and

y=d

.

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easy

Find the area enclosed between the curve

y = \sin x

and the x-axis from

x=0

to

x=\pi

.

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easy

State the formula for the area of the region bounded by the curve

y=f(x)

, the x-axis, and the lines

x=a

and

x=b

.

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medium

Calculate the area of the region enclosed between the two parabolas

y^2 = 4x

and

x^2 = 4y

.

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medium

Explain in detail the concept of finding the area under a curve

y = f(x)

from

x=a

to

x=b

as the limit of a sum. Use a diagram to support your explanation.

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medium

Sketch the graph of

y = |x-2|

and use integration to calculate the area bounded by the curve, the x-axis, and the lines

x=0

and

x=4

.

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medium

Analyze the curve

y = x^2 - 4

and calculate the area of the region bounded by it and the x-axis.

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medium

Compare the area bounded by the curve

y=x^3

, the x-axis, from

x=0

to

x=2

with the area of the rectangle formed by the lines

x=0, x=2, y=0

, and

y=8

.

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medium

Find the area of the region bounded by the parabola

y=x^2

and the line

y=4

.

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medium

Write the definite integral that represents the area of the region in the first quadrant bounded by the parabola

y^2 = 8x

, the x-axis, and the line

x=2

. Do not evaluate the integral.

Try solving it in your notebook first, then check the solution.

medium

Evaluate the claim: 'To find the area between

y=f(x)

and

y=g(x)

, the integral is always

\int_{a}^{b} [f(x) - g(x)] \,dx

.'

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medium

Summarize the approach to calculate the area bounded by a curve

y=f(x)

, the x-axis, and ordinates

x=a

and

x=b

, when the curve crosses the x-axis at a point

x=c

where

a < c < b

. Use a diagram to illustrate your explanation.

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medium

Justify why the area bounded by the curve

y = \cos x

, the x-axis, from

x=0

to

x=\pi

is not given by the single integral

\int_{0}^{\pi} \cos x \,dx

. Formulate the correct expression.

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medium

Using integration, justify that the area of the region bounded by the ellipse

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

is

\pi ab

square units. You must derive this result from a definite integral.

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medium

Summarize the key steps required to find the area bounded by the curve

y=x^2

, the x-axis, and the ordinates

x=1

and

x=3

.

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medium

Under what condition will the definite integral

\int_{a}^{b} f(x) \, dx

yield a negative value?

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medium

Formulate and evaluate the integral to find the area of the region bounded by the curve

y = |x-1|

and the line

y=3

.

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medium

Calculate the area of the region bounded by the parabola

y^2 = 8x

and its latus rectum.

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medium

Solve for the area under the curve

y = e^x

from

x=0

to

x=1

.

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medium

A student sets up the integral

\int_{0}^{4} (\sqrt{16-y^2}) \,dy

to find the area bounded by the circle

x^2+y^2=16

in the first quadrant. Critique this setup. Is it correct? Justify your answer.

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medium

Set up the definite integral required to calculate the total area of the ellipse

\frac{x^2}{25} + \frac{y^2}{16} = 1

. You do not need to evaluate it.

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medium

Explain the method used to find the total area for a curve

y=f(x)

between

x=a

and

x=b

, if the curve lies partially above and partially below the x-axis.

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medium

Justify the choice of horizontal strips over vertical strips for finding the area of the region bounded by the parabola

x=y^2

and the line

x=y+2

. Then, calculate the area.

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medium

Describe what the expression

dA = y \, dx

represents when calculating the area under a curve.

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hard

Create a non-zero function

f(x)

such that the value of the definite integral

\int_{-1}^{1} f(x) \,dx

is zero, but the area bounded by

y=f(x)

, the x-axis,

x=-1

and

x=1

is not zero. Justify your choice.

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hard

Calculate the area of the circle described by the equation

x^2 + y^2 - 6x = 0

using integration.

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hard

Formulate and evaluate an integral to find the area bounded by the loop of the curve

y^2 = x(x-1)^2

.

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hard

The area bounded by the parabola

y^2=8x

and the line

x=2

is divided into two equal regions by the line

x=c

. Formulate the equation involving integrals that must be solved to find the value of

c

, and then solve for

c

.

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hard

Using integration, find the area of the triangular region with vertices at A(-1, 0), B(1, 3), and C(3, 2).

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hard

Evaluate the area of the region defined by the inequalities

\{(x, y) : 0 \le y \le x^2+1, 0 \le y \le x+1, 0 \le x \le 2\}

.

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hard

Explain the fundamental difference between using vertical strips (integrating with respect to

x

) and horizontal strips (integrating with respect to

y

) to find the area of a region.

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hard

Calculate the area of the smaller region bounded by the circle

x^2 + y^2 = 4

and the line

x=1

.

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hard

Explain why the absolute value must be taken when calculating the area of a region that lies entirely below the x-axis. Illustrate your explanation with a general function

f(x)

where

f(x) < 0

for all

x

in the interval

[a, b]

.

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hard

Describe the complete method to find the total area enclosed by the ellipse

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

using integration with respect to the y-axis (i.e., using horizontal strips).

Try solving it in your notebook first, then check the solution.

Application of IntegralsClass 12 Mathematics Practice Questions