f(x) g(x) ∫(f−g)dx
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Chapter 8  ·  Class XII Mathematics  ·  MCQ Practice

MCQ Practice Arena

Application of Integrals

From Curves to Area — Visualize the Region Before You Integrate

📋 50 MCQs ⭐ 12 PYQs ⏱ 90 sec/Q
🚀 Start Practising 📖 Revise Notes First

MCQ Bank Snapshot

50Total MCQs
15Easy
20Medium
15Hard
12PYQs
90 secAvg Time/Q
10Topics
Easy 30% Medium 40% Hard 30%

Why Practise These MCQs?

JEE MainCBSE

This short but scoring chapter turns the previous chapter's techniques into region-finding questions. CBSE boards favour simple curve-and-line area problems, while JEE Main includes intersecting-curve MCQs that reward a quick sketch over a long calculation. Getting the region right on paper is worth more than any formula here.

Topic-wise MCQ Breakdown

Area under Simple Curves28 Q
Area between Two Curves22 Q

Must-Know Formulae Before You Start

Recall these cold before attempting MCQs — they appear in >70% of questions.

$\text{Area} = \int_a^b y\,dx$
$\text{Area} = \int_a^b (f(x)-g(x))\,dx$
$\text{Area (w.r.t. } y\text{-axis)} = \int_a^b x\,dy$

MCQ Solving Strategy

Always sketch the curve (or curves) before writing a single integral — the sketch tells you the correct limits and which function sits on top. If two curves intersect inside the given interval, split the area into separate integrals at the intersection point rather than integrating one combined expression across the whole range.

⚠ Common Traps & Errors

Difficulty Ladder

Work through each rung in order — do not jump to Hard before mastering Easy.

① Easy

Area under a single curve with straightforward limits

② Medium

Area between a curve and a line, region bounded by simple intersections

③ Hard

Area between two curves with multiple intersection points

★ PYQ

JEE Main — intersecting-curve area with limit-finding; CBSE — sketch-based region area

Continue Your Preparation

📖 Chapter Notes Revise concepts & theory 🧠 MCQ Practice Topic-wise Questions 🎯 PYQ Bank Previous year questions ✔️❌ True / False Concept-based True & False questions 📘✔️ Exercises Step by step Solutions to textbook exercises ➡ Next Chapter Differential Equations MCQs
🎯 Knowledge Check

Mathematics — APPLICATION OF INTEGRALS

50 Questions Class 12 MCQs
1
The area bounded by the curve \(y=x\), the x-axis, and the ordinates \(x=0\) and \(x=2\) is:
2
The area under the curve \(y=x^2\) between \(x=0\) and \(x=1\) is:
3
The area bounded by \(y=4\), the x-axis, and the lines \(x=0\) and \(x=3\) is:
4
The area under the curve \(y=\sin x\) from \(x=0\) to \(x=\pi\) is:
5
The area bounded by \(y=\cos x\), the x-axis, and the ordinates \(x=0\) and \(x=\frac{\pi}{2}\) is:
6
The geometrical area between the line \(y=-x\), the x-axis, and the ordinates \(x=0\) and \(x=2\) is:
7
The area under the curve \(y=\sqrt{x}\) between \(x=0\) and \(x=4\) is:
8
The area under the curve \(y=1-x\) above the x-axis between \(x=0\) and \(x=1\) is:
9
The area bounded by the line \(y=2x+1\), the x-axis, and the ordinates \(x=0\) and \(x=1\) is:
10
The area enclosed between the curves \(y=x\) and \(y=x^2\) in the interval \(0\leq x\leq1\) is:
11
The area enclosed by the parabola \(y=4-x^2\) and the x-axis is:
12
The points of intersection of the curves \(y=2x\) and \(y=x^2\) are:
13
The area enclosed by the lines \(y=x\), \(y=2x\), and \(x=1\) in the first quadrant is:
14
The area bounded by the curve \(y=|x|\), the x-axis, and the ordinates \(x=-1\) and \(x=1\) is:
15
The area enclosed by the upper semicircle \(x^2+y^2=4\) and the x-axis is:
16
The area under the straight line \(y=2x+2\) between \(x=-1\) and \(x=1\) is:
17
The area enclosed by the parabola \(y=4x-x^2\) and the x-axis is:
18
The area enclosed between \(y=\sqrt{x}\) and \(y=x\) in the first quadrant is:
19
The area under the curve \(y=\sin x\) between \(x=0\) and \(x=\frac{\pi}{2}\) is:
20
The area under the curve \(y=\frac{1}{x}\) between \(x=1\) and \(x=e\) is:
21
The area enclosed between the curves \(y=x^2\) and \(y=2-x\) is:
22
The area enclosed between the parabolas \(y=x^2\) and \(y=2-x^2\) is:
23
The area enclosed by the circle \(x^2+y^2=9\) in the first quadrant is:
24
The area enclosed by the parabola \(x=y^2\) and the line \(x=2\) is:
25
The area enclosed between the curves \(x=y^2\) and \(x=y\) is:
26
The area under the curve \(y=3x^2\) between \(x=-1\) and \(x=1\) is:
27
The area enclosed by the parabola \(y=1-x^2\) and the x-axis is:
28
The area enclosed between \(y=|x|\) and \(y=1\) is:
29
The area bounded by \(y=\cos x\) and the x-axis from \(x=-\frac{\pi}{2}\) to \(x=\frac{\pi}{2}\) is:
30
The area enclosed between the curves \(y=x^3\) and \(y=x\) from \(x=-1\) to \(x=1\) is:
31
The area enclosed between the line \(y=4\) and the parabola \(y=x^2\) in the first quadrant is:
32
The area enclosed by the line \(x+y=4\) and the coordinate axes is:
33
The area enclosed between the curves \(y=2\) and \(y=x\), and the y-axis, is:
34
The area under the curve \(y=e^x\) between \(x=0\) and \(x=\ln 2\) is:
35
The area under the curve \(y=\frac{1}{1+x^2}\) between \(x=-1\) and \(x=1\) is:
36
The area enclosed between the parabolas \(y=x^2\) and \(y=4x-x^2\) is:
37
The area enclosed between the parabola \(y^2=4x\) and the line \(x=4\) is:
38
The area enclosed between the parabola \(y^2=4x\) and the line \(x=y\) is:
39
(PYQ-pattern) The area enclosed by the curve \(y=\sqrt{4-x^2}\), the x-axis and the y-axis is:
40
(PYQ-pattern) The area enclosed by the parabola \(y=x^2-4\) and the x-axis is:
41
(PYQ-pattern) The area enclosed between the curves \(y=x^2\) and \(y=|x|\) is:
42
(PYQ-pattern) The area enclosed between the curves \(y=x^2\) and \(y=2|x|\) is:
43
(PYQ-pattern) The area enclosed between the curves \(y=\sin x\) and \(y=\cos x\) from \(x=0\) to \(x=\frac{\pi}{2}\) is:
44
(PYQ-pattern) The area enclosed between \(y=x\) and \(y=\frac{1}{x}\) between \(x=1\) and \(x=2\) is:
45
(PYQ-pattern) The area enclosed between the parabolas \(y=x^2\) and \(y=4-x^2\) is:
46
(PYQ-pattern) The area enclosed between the curves \(y=x^2\) and \(y=4-3x\) is:
47
(PYQ-pattern) The area enclosed between the curves \(y=x^2\) and \(y=x^4\) from \(x=-1\) to \(x=1\) is:
48
(PYQ-pattern) The area enclosed between the curves \(y=\sqrt{x}\) and \(y=x^2\) in the first quadrant is:
49
(PYQ-pattern) The area enclosed between the parabola \(y=x^2\) and the line \(y=1\) is:
50
(PYQ-pattern) The total area enclosed between the curves \(y=\sin x\) and \(y=\sin 2x\) from \(x=0\) to \(x=\pi\) is:
📚
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Class 12 Maths Chapter 8: Application of Integrals MCQs
Class 12 Maths Chapter 8: Application of Integrals MCQs — Complete Notes & Solutions · academia-aeternum.com
Prepare effectively for your CBSE Class 12 Mathematics examination with our comprehensive collection of Application of Integrals MCQs, designed according to the NCERT syllabus. This practice set covers important concepts such as the geometrical interpretation of definite integrals, area under curves, areas bounded by two curves, points of intersection, and areas enclosed by circles and parabolas. The 50 multiple-choice questions are arranged in increasing order of difficulty, progressing from…
🎓 Class 12 📐 Mathematics 📖 NCERT ✅ Free Access 🏆 CBSE · JEE
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Frequently Asked Questions

Application of Integrals is a chapter that explains how definite integrals are used to calculate the areas of geometrical regions bounded by curves, straight lines and coordinate axes.

The chapter covers the geometrical interpretation of definite integrals, area under a curve, area bounded by curves, and the calculation of areas between two curves.

A definite integral represents the signed area between the graph of a function and the x-axis over a specified interval. Areas above the x-axis are positive, while areas below it are negative.

The area under a curve y = f(x), above the x-axis, between x = a and x = b is calculated using the definite integral ??? f(x) dx, provided f(x) is non-negative throughout the interval.

The area between two curves y = f(x) and y = g(x), where f(x) is the upper curve, is calculated using ??? [f(x) - g(x)] dx over the given interval.

A definite integral gives the signed area, accounting for positive and negative contributions, whereas geometrical area is always non-negative. Therefore, absolute values or suitable splitting of intervals may be required.

The area can be calculated by integrating with respect to y, using the formula A = ??? [x_right - x_left] dy, where x_right and x_left represent the right and left boundaries.

Integration with respect to x (dy/dx representation) is generally used when vertical strips simplify the region, while integration with respect to y (dx/dy representation) is preferred when horizontal strips provide simpler limits and boundaries.

Points of intersection determine the limits of integration and help identify which curve lies above or below the other, ensuring that the correct bounded region is considered.

Application of Integrals is important for CBSE Board Examinations and competitive examinations such as JEE and CUET. Understanding definite integrals, graphical interpretation, intersection points and area calculations helps students solve both conceptual and application-based problems.

This practice set contains 50 multiple-choice questions arranged in increasing order of difficulty, covering fundamental concepts and advanced applications of integrals.

The MCQs cover geometrical interpretation of definite integrals, area under curves, area between two curves, areas bounded by circles and parabolas, points of intersection, symmetry, and integration with respect to x and y.

Yes, the questions are designed around the concepts and applications prescribed in NCERT Class 12 Mathematics Chapter 8.

Yes, every question includes four options, the correct answer, and a concise explanation to help students understand the mathematical concepts and calculations.

The question bank includes easy, medium, and hard questions, progressing from basic area calculations to challenging problems involving multiple curves and definite integrals.

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